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DIFFERENTIAL    CALCULUS. 


THE 


DIFFERENTIAL  CALCULUS : 


UNUSUAL  AND  PARTICULAR  ANALYSIS  OF  ITS  ELEMENTARY 

i 

PRINCIPLES,  AND  COPIOUS  ILLUSTRATIONS  OP 
ITS  PRACTICAL  APPLICATION. 


JOHN    SPARE,   A.  M.,M.D. 


BOSTON: 
BRADLEY,   DAYTON   AND   COMPANY, 

20  WASHINGTON   STREET. 
1865. 


Entered,  according  to  Act  of  Congress,  in  the  year  1865,  by 

JOHN    SPARE, 
In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


In  manuscript,  by  the  same  author,  and  the  publication  contemplated, 

THE   ELEMENTS   OF 
DEMONSTRATIVE   GENERAL  ARITHMETIC. 


ELECTROTYPED    AT    THK 
BOSTON   STEREOTYPE    FOUNDRY, 

No.  U  Spring  Lane. 


OF  QEO.   C.   RAND  AND  AVEEY. 


PREFACE. 


THE  present  treatise  on  the  Differential  Calculus  is 
believed  to  be  the  first,  of  any  character,  that  has  been  written 
and  published  in  America  as  the  special  topic  of  a  volume ; 
and  the  first,  so  far  as  known  to  the  author,  ever  published, 
that  professes  the  character  of  the  present  one. 

In  our  country,  and  at  this  day,  every  mathematical  book 
must  be  eminently  analytical  and  practical  for  the  many.  In 
European  countries,  and  past  times,  works  upon  the  Differen 
tial  Calculus  have  labored  to  show,  with  curtness  and  severity, 
just  what  the  science  has,  at  any  date,  discovered  in  far- 
reaching  achievement,  for  the  few.  Some  compilations  of 
the  latter  have  been  adopted  here,  to  find  which,  it  is 
necessary,  for  the  most  part,  to  search  works  on  analytical 
geometry. 

The  calculus  being  algebra,  a  strictly  numerical  science, 
the  present  treatise  claims  to  have  labored  successfully  in 
putting  on  the  true  character  as  such.  No  insinuation  is 

(v) 

740488 


Vi  .     .       PIIEFACE. 

allowed  to  prevail  that -it  is  any  part  whatever  of  analytical 
rg^otnibttyy* or t  that"  it 'is- -other  than  the  natural  sequel  and 
supplement  of  common  algebra ;  useful,  indeed,  as  an  appli 
ance,  to  borrow,  in  investigation  of  the  few  kinds  of  geomet 
rical  quantity. 

Aware  of  the  indispensable  importance,  to  a  learner  of  any 
new  branch  of  science,  that  his  already  acquired  knowledge 
of  the  most  nearly  allied  character  should  be  adopted  as  the 
central  principle,  around  which  the  new  ideas  and  sugges 
tions  are  to  acquire  distinctness  and  character,  the  author  has 
commenced  this  treatise  with  the  terms  and  appliances  of 
algebra,  assiduously  preserved  and  employed.  The  student  is 
thus  enabled  to  hold  his  familiar  ground,  see  his  former  paths 
and  landmarks,  find  the  new  objects  designed  for  his  atten 
tion,  tangible  and  actual,  the  fruits  such  that  he  may  grasp 
them,  and  add  to  the  previous  nurture  and  furnishings  of  his 
mind.  Accordingly,  he  will  find  here  his  favorite  algebraic 
problems  placed  before  him  in  the  phase  in  which  the  calculus 
is  required  for  their  solution.  The  author  is  not  aware  that 
concrete,  practical  problems  of  this  character  were  ever  before 
published.  In  this  manner  is  shown  the  early  and  element 
ary  nature  of  the  calculus  ;  that  it  entwines  itself  around  the 
very  threshold  of  mathematical  inquiry. 

It  is  a  definite,  but  perfectly  normal  fact  in  the  history  of 
science,  that  the  distinguished  explorers  of  the  mathematical 


PREFACE.  VU 

laws  of  physical  science  were  obliged  to  suspend  their 
researches,  and  come  to  a  stand-still,  in  order  to  develop, 
for  necessary  use,  the  elementary  principles  of  the  calculus, 
then  unknown  to  the  world.  Thus  always  will  a  neglect  of 
these,  or  any  elementary  truths,  by  persons  who  should  be 
well  informed  concerning  them,  avenge  itself  afterwards  in 
their  perplexity. 

This  treatise  has  been  prepared  under  the  strong  conviction 
that  its  plan  of  analysis  could  not  fail  of  adoption  at  some 
stage  of  the  natural  prosecution  of  our  general  modern  system 
of  instruction :  it  is  simply  the  extension  to  this  science, 
in  which  an  interest  is  becoming  manifest,  of  just  the  ana 
lytical  methods  already  in  use  in  reference  to  most  other 
branches  of  learning,  particularly  elementary  mathematics. 

In  the  many  practical  problems  offered  for  solution,  as  a 
distinguishing  feature,  the  work  aims  at  cultivating  and 
prolonging  the  enthusiasm  of  the  student,  by  clothing  his 
conceptions  of  quantity  in  the  garb  of  romance,  or  something 
of  a  supposable  human  experience  ;  these  conceptions  may, 
with  the  more  interest,  be  erratic  and  fanciful  as  to  econom 
ical  life,  without  ever  filling  or  exhausting  the  generality  of 
pure  mathematical  conception. 

It  would  not  be  practicable  to  present  in  detail,  in  this 
place,  the  different  features  of  the  work.  It  has  been  pre 
pared  with  a  great  deal  of  pains,  and  with  reference  to  a 


Vlll  PREFACE. 

well-considered  plan.  The  consecutiveness  of  the  analysis 
has  been  kept  in  view ;  not  the  accommodation  of  the  equal 
and  consistent  progress  by  a  learner  of  a  given  intelligence, 
through  all  its  pages,  in  a  given  number  of  weeks.  It  has 
rather  the  character  of  a  hand-book,  for  collateral  use 
through  much  of  a  mathematical  course,  meeting  the  differ 
ent  grades  of  the  growing  intelligence  of  a  few  years,  as  to 
one  person. 


NEW  BEDFORD,  MASS.,  1865. 


CONTENTS. 


SECTION   I. 

ELEMENTARY    PRINCIPLES.  — A    VARIABLE. 

Page 

The  purpose  of  the  Differential  Calculus,          ....  1 

Historical  statement  in  regard  to  the  Calculus,     ...  1 

Example  of  a  variable  quantity, 2 

Determinate  relative  values,          .         .        .         .         .        .  3 

Indeterminate  quantities  in  an  equation  of  the  first  degree 

having  two  variables,         .......  4 

Instance  of  a  function  of  a  variable,     .....  5 

A  variable,  how  represented,    .......  5 

Independent  and  dependent  variables, 5 

On  changing  a  variable  with  reference  to  a  change  of  investi 
gation,      ..........  6 

The  possible   permutations   of  variables   among  conditioned 

quantities,     .........  6 

Comprehensive  enunciation  of  a  problem,  with  the  possible 

permutations  of  two  variables, 7 

SECTION    II. 

DEFINITIONS  RELATING  TO  FUNCTIONS.  — THE  USE  OF  SIGNS. 

An  explicit  algebraic  function  defined, 8 

The  value  of  a  function, 8 

(ix) 


X  CONTENTS. 

Page 

Functions  which  are  not  algebraic, 8 

An  implicit  function  of  a  variable,   ......       9 

A  formula,   ..........  9 

The  relation  of  a  function  of  a  single  variable  to  an  equation,       9 
Numerical  nature  of  all  functions,         .         .         .         .         .         10 

The  signs  in  use,      .         .         .         .         .         .         .         .         .11 

Elementary  instance  of  the  nature  of  the  Binomial  Theorem,     11 
Fluents  and  fluxions  mentioned  as  obsolete  terms,  .        .         .12 


SECTION    III. 

THE  DEGREES  OF  AN  EQUATION.  —  FIRST  AND  SECOND 
DEGREES. 

General  form  of  an  equation  of  the  First  Degree  having  two 

variables, 13 

Theorem  relating  to  the  general  equation  of  the  First  Degree 

having  two  variables, 14 

General  equation  of  the  Second  Degree  having  two  variables,     15 
Two  variables  as  related  to  the  fundamental  arithmetical  rules,     15 
Rule  for  determining  the  degree  of  an  equation,  .         .         .         16 
Classification  of  certain  systems  of  the  relative  values  of  the 
two   variables   in   the    general   equation   of  the    Second 
Degree, 17 

SECTION    IV. 

INCREASE  OF  THE  VALUE  OF  FUNCTIONS.  — DECREASE  OF  THE 
VALUE  OF  FUNCTIONS.  —  STATIONARY  VALUES  OF  THEM. 

Exercises, 1$ 

The  purpose  of  an  increment  and  decrement  of  a  variable,      .     20 
Minuendive  and  subtrahendive  terms  of  a  function,      .         .         21 


CONTENTS.  XI 

Page 
Elementary  notions  of  the  maximum  and  minimum  values  of 

certain  simple  forms  of  functions,  ....  22 
Practical  examples  of  the  same,  with  solutions  by  the  use  of 

inequations,       .........     22 

SECTION    V. 

THE  BINOMIAL  SERIES.— SOLUTIONS   OF   PROBLEMS  BY 
INEQUATIONS. 

Theorem :  how  any  term  of  this  series  is  made  greater  or  less 

than  the  sum  of  all  following  terms,  ....  27 

Solutions  of  problems  by  inequations  when  developments  give 

rise  to  a  protracted  series,  ......  28 

SECTION    VI. 

DIFFERENTIAL  OF  A   VARIABLE.  —  DIFFERENTIAL  OF  A 
FUNCTION. 

Discovered  use  for  the  second  term  of  the  binomial  series, 
when  a  function  is  developed  by  it.  on  the  variable  taking 
an  increment,         ........         35 

The  set  of  sub-terms  that  may  compose  the  second  one  in  such 

development,    .........     35 

Differential  of  a  function, 35 

Differential  of  a  variable, 35 

Symbols  for  representing  a  function  of  a  single  variable,  and 

the  differential  of  a  single  variable,  ....  35 
Rule  for  differentiating  a  function  when  it  is  any  entire  power 

of  the  variable,  having  or  not  having  a  constant  factor,    .     36 
Rule,  when  the  function  has  a  term  that  is  a  constant,          .         36 
Rule,  when  the  function  is  a  product  of  certain  other  func 
tions  of  the  variable, 37 


Xll  CONTENTS. 

Page 
Rule,  when  the  function  is  a  sum  or  difference  of  certain  other 

functions  of  the  variable, 38 

Rule,  when  the  function  is  a  fraction, 38 

Rule,  when  the  function  has  a  fractional  exponent,       .        .        41 

SECTION    VII. 

FIRST  DIFFERENTIAL  COEFFICIENT. 

First  differential  coefficient, 42 

Ratio  of  the  differential  of  a  function  to  that  of  its  single 

variable, 43 

Examples  of  differential  coefficients, 44 

Reciprocal  of  a  differential  coefficient,          ....  46 

SECTION    VIII. 

USE  OF  FIRST  DIFFERENTIAL  COEFFICIENT. 

The  value  of  a  differential  coefficient,       .        .         .        .         .48 
Problems  illustrating  the  practical  signification  of  a  differential 

coefficient, 50 

SECTION    IX. 


SUCCESSIVE  DIFFERENTIATION.  — SECOND,  THIRD,  ETC., 
DIFFERENTIALS. 

A  first  differential  coefficient  may  be  a  variable  function, .  .  56 
The  differentiation  of  such  derived  function,  ...  57 
The  notation  deduced  for  successive  differentiations,  .  .  58 


CONTENTS.  Xlll 

Page 
The  law  of  successive  differences  of  the  arithmetical  powers  of 

the  natural  series  of  numbers, 60 

On  what  the  sign  of  any  of  the  successive  differential  coeffi 
cients  depends, 61 

SECTION    X. 

\ 

TAYLOR'S  THEOREM. 

The  purpose  of  Taylor's  Theorem, 61 

On  developing  a  function  of  a  single  variable  when  the  vari 
able  takes  an  increment  or  decrement,  by  this  theorem,  in 

examples, 62 

Importance   of  Taylor's   Theorem,  —  testimony   of  Professor 

Playfair, 64 

SECTION    XI. 

THEORY  OF  MAXIMA  AND  MINIMA. 

The  proper  definitions  of  maxima  and  minima,         .         .         .65 
Demonstration  of  the  theory  of  maxima  and  minima  by  Tay 
lor's  Theorem, 67 

Enunciation  of  the  principle,    .......     69 

On  abridging  the  method,     .        .        .        .        .        .        .        70 

Problems  in  maxima  and  minima, 72 

SECTION    XII. 

PROBLEMS  FURNISHING  EXPLICIT  FUNCTIONS    OF  ONE  VARI 
ABLE;    FOR  DETERMINING   THEIR  MAXIMA  AND  MINIMA. 

Algebraic  results  which  can  have  no  practical  application  as 

answers  to  problems,     .......        75 

Problems,         . 75 

b 


XIV  CONTENTS. 

SECTION    XIII. 

COMPLETE  HISTORY  OF  FUNCTIONS. 

Page 

Suggestions  relating  to  a  complete  investigation  of  the  range 
of  the  values  of  a  function,  with  reference  to  those  of  the 
variable,  .........  96 

Illustration  by  an  example, 98 

SECTION    XIV. 

PRINCIPLES  AND  PROBLEMS  RELATING  TO  PROJECTED  BODIES. 

Formula  or  function  defining  the  course  of  a  projected  body,  101 
Problems  relating  to  projected  bodies, 105 


SECTION    XV. 

THE  SYSTEM  OF  SYMBOLS  FOR  FUNCTIONS.  —  AN  IMPLICIT 
FUNCTION  OF  A  SINGLE  INDEPENDENT  VARIABLE  AND  ITS 
DIFFERENTIATION. 

The  symbols, 108 

General  notation  for  a  function  of  more  than  one  variable,        .  109 
The  relation  of  functions  to  algebraic  equations  with  two  or 

more  unknown  quantities, HO 

Differentiation  for  functions  of  (x,  y)  =  0,      .         •         •         .111 

A  rule  for  the  same, 1*3 

Contradictory  equations, H5 

Diverse  values  of  0, H6 

Differentials  of  any  two  functions  of  a  single  variable  not  ne 
cessarily  equal  in  value,  although  the  functions  may  be,    .  117 


CONTENTS.  XV 


SECTION    XVI. 

PROBLEMS  WHICH  MAY  FURNISH  IMPLICIT  FUNCTIONS  OF 
ONE  VARIABLE,  AND  CASES  OF  THEIR  MAXIMA  AND 
MINIMA. 

Page 

The  problems,      .        . 118 


SECTION    XVII. 

FUNCTIONS  OF    TWO    INDEPENDENT    VARIABLES:    THEIR   DIF 
FERENTIATION  AND  THEIR  MAXIMA  AND   MINIMA. 

Mode  of  differentiating, 130 

Development  by  Taylor's  Theorem  of  /  (x  ±  h,  y  ±  k),        .       132 
Theory  for  f  (x,  y)  =z  z  at  a  maximum  or  minimum,        .         .  133 


SECTION    XVIII. 

PROBLEMS  RELATING  TO  FUNCTIONS   OF    TWO  INDEPENDENT 
VARIABLES,  AND  CASES  OF  THEIR  MAXIMA  AND  MINIMA. 

The  problems, 136 


SECTION    XIX. 

DEMONSTRATION  OF  THE  GENERAL  FORM  OF  THE  DEVELOP 
MENT  OF  /  (x  +  ft),  AND  OF  THE  DIFFERENTIATION  OF  CER 
TAIN  FUNCTIONS. 

The  demonstration, 143 

The  development  formally  applied  in  the  demonstration  of  the 

differentiation  of  the  elementary  forms  of  functions,      .       147 


XVI  CONTENTS. 

SECTION    XX. 

MACLAURIN'S  THEOREM,  AND  ITS  APPLICATION. 

Page 

Its  form  and  demonstration, 150 

The  theorem  derived  also  from  Taylor's,       .        .        .        .151 

Binomial  Theorem  demonstrated, 154 

Maclaurin's  Theorem  applied  in  cases  off  (x,  y}  =:0, .         .       156 
Failure  of  Maclaurin's  Theorem,      .        .        .        .        .        .159 

Maclaurin's  Theorem  applied  in  cases  of  F  (x,  y)  :=  2,  .        .       160 

SECTION    XXI. 

DETERMINATION  OF  THE  VALUE  OF  VANISHING    FRACTIONS. 

Nature  of  a  Vanishing  Fraction, 160 

Demonstration  of  the  method  of  determining  the  value  of  a 

Vanishing  Fraction,  .  .  .  .  .  .  .164 

The  value  of  a  fraction  of  which  both  the  numerator  and  de 
nominator  become  infinite, 166 

Value  of  the  difference  of  two  functions  when  each  becomes 

infinite, ..........  167 

Infinite  quantities, 167 

SECTION    XXII. 

EXCEPTIONAL  PRINCIPLE  RELATING  TO  TAYLOR'S  THEOREM. 

When  Taylor's  Theorem  fails, 168 

Cases  of  maxima  and  minima  when  this  theorem  fails,  .  .169 
True  development  when  this  theorem  fails,  .  .  .  .170 
Value  of  vanishing  fractions  in  such  cases,  .  .  .  .  .172 


CONTENTS.  XVii 


SECTION   XXIII. 

NATURE  OF  LOGARITHMS  AND  EXPONENTIAL  QUANTITIES. 

Page 

A  Logarithm, 174 

The  Common  System, 174 

Mode  of  expressing  Logarithms,          .        .         .        .        .175 
Operations  by  the  use  of  Logarithms, 1 76 


SECTION    XXIV. 

DIFFERENTIATION  AND  DEVELOPMENT  OF  LOGARITHMIC  AND 
EXPONENTIAL  FUNCTIONS. 

These  functions  defined, 178 

Rule  for  differentiating  a  Logarithmic  Function,      .        .        .  180 
Rule  for  Exponential  Functions,  ......       182 

Base  of  the  hyperbolic  system  deduced,   .         .         .         .         .183 

Modulus  of  the  common  system  deduced,     ....       183 

Mode  of  designating  certain  logarithms, 185 


SECTION   XXV. 

EXAMPLES    OF    THE    DIFFERENTIATION    AND    DEVELOPMENT 
OF  LOGARITHMIC  FUNCTIONS. 

The  calculation  of  logarithms, 186       ^A 

Examples  of  differentiation, 189 

The  differentiation  of  certain  algebraic  functions  facilitated  by 

logarithms,    .        . 190 

The  determination  of  maxima  and  minima  facilitated  by  loga 
rithms,      191 

b* 


XVlll  CONTENTS. 


SECTION    XXVI. 

EXAMPLES  OF  THE  DIFFERENTIATION  AND  ANALYSIS  OF  EX 
PONENTIAL  FUNCTIONS  ;  INCLUDING  EXAMPLES  FROM  COM 
POUND  INTEREST  AND  INCREASE  OF  POPULATION. 

Page 
The  examples, 193 

The  series  called  progression  by  quotient,         ....  196 

Compound  interest,       ........       198 

Maxima  and  minima  of  exponential  functions,  .  .  .  201 
Interest  of  money  for  the  successive  months  of  a  year,  .  204 
True  compound  interest  for  years  with  fractional  parts  of  a 

year, 205 

SECTION    XXVII. 

DIFFERENTIATION  OF  CIRCULAR  FUNCTIONS. 

Their  definition  and  mode  of  expression,  ....  206 
Differentiation  of  sin.  x,  cos.  a;,  etc.,  .....  208 
Development  of  these  functions  by  Maclaurin's  Theorem,  .  210 
Calculation  of  the  natural  sine,  etc.,  of  an  arc,  .  .  .214 
Maxima  and  minima  of  Circular  Functions  .  .  .  .216 
Vanishing  Fractions  with  Circular  Functions,  .  .  .  .216 

SECTION    XXVIII. 

GEOMETRICAL  ILLUSTRATIONS  OF  THE  VALUES  OF  FUNC 
TIONS,  AND  THE  CORRESPONDING  VALUES  OF  THEIR  VARI 
ABLES  ;  ALSO  OF  THE  VALUE  OF  DIFFERENTIAL  COEFFI 
CIENTS,  MAXIMA  AND  MINIMA,  ETC. 

The  possible  linear  value  of  algebraic  expressions,  .  .  217 
The  linear  construction  of  the  values  of  a  function  for  one  or 

several  assumed  values  of  the  variable,     ....  218 


CONTENTS.  XIX 

Page 

Construction  of  equations  of  the  first  degree,      .        .        .      220 
Construction  for  algebraic  problems  in  equations  of  the  first 

degree, 221 

The  arithmetical  rule  of  Double  Position  illustrated  by  con 
struction,       222 

First  and  succeeding  differential  coefficients  illustrated  by  con 
struction,  223 

Maxima  and  minima  illustrated  by  construction,  .         .         .       225  ' 
Construction  of  the  general  equation  of  the  second  degree 

having  two  variables, 225 

Examples  of  construction  within  the  second  degree,     .         .       234 
More  general  examples  of  construction   than  the  first  two 

degrees, 235 

Construction  for  illustrating  the  value  of  Vanishing  Fractions,  237 
Construction  of  the  value  of  logarithms  both  for  the  common 

and  the  hyperbolic  systems,       ......  238 

Construction  for  functions  of  two  independent  variables,      .      240 
Critical  remarks, 242 


DIFFERENTIAL  CALCULUS. 


SECTION  I. 

ELEMENTARY  PRINCIPLES.— A  VARIABLE. 

1.  THE  mode  of  developing  the  nature  of  the  Differen 
tial  Calculus  to  be  adopted  in  this  treatise  will  be,  taking  a 
point  of  departure  within  the  common  principles  of  alge 
bra  ;  for  it  is  within  these  principles  that  the  Calculus  is 
based;  so  far  as  algebraic  quantities  are  concerned,  the 
calculus  is  but  the  completion  of  omitted  algebraic  princi 
ples  —  omitted  from  the  general  consideration  of  algebra, 
for  historical  reasons  only  —  later  and  separate  invention. 

Algebra,  in  its  ordinary  character,  had  omitted  to  deter 
mine  a  System  of  Principles  according  to  which  the  value 
or  resulting  amount  of  a  formula  must  be  inferred  to  change, 
when  a  particular  component  quantity  or  quantities  within 
that  formula  should  be  supposed  to  increase  or  to  decrease, 
when  near  any  specific  value,  or  while  passing  through  a 
range  of  all  possible  values.  This  is  the  purpose  of  the 
Differential  Calculus. 

2.  The  following  Historical  Statement  in  regard  to  the 
Calculus  has  been  derived  from  Professor  Playfair's  Dis 
sertation  on  the  Progress  of  Mathematical  and  Physical 
Science : 

"  Of  the  new  or  infinitesimal  analysis,  we  are  to  con 
sider  Sir  Isaac  Newton  as  the  first  inventor,  Leibnitz,  a 
1  (1) 


2  DIFFERENTIAL  CALCULUS. 

German  philosopher,  as  the  second ;  the  latter's  discovery, 
though  posterior  in  time,  having  been  made  independently 
of  the  former's,  and  having  no  less  claim  to  originality. 
It  [the  latter's]  had  the  advantage  also  of  being  first  made 
known  to  the  world,  which  was  in  1684. 

"  This  infinitesimal  analysis,  the  greatest  discovery  ever 
made  in  the  mathematical  sciences,  as  it  became  known 
every  where  enlarged  the  views,  roused  the  activity  and 
increased  the  power  of  geometers,  while  it  directed  their 
warmest  sentiments  of  gratitude  and  admiration  towards  the 
great  inventors.  By  its  introduction  the  domain  of  the 
Mathematical  Sciences  was  incredibly  enlarged  in  every 
direction.  Although  developed  in  a  state  applied  to  geome 
try,  it  was  afterwards  justly  inferred  to  be  independent  of  it. 

"  The  fluxionary  and  differential  calculus  may  be  con 
sidered  two  modifications  [in  the  matter  of  notation]  of  one 
general  method,  aptly  distinguished  by  the  name  of  the  in 
finitesimal  analysis." 

3t  In  preference  to  the  use  of  abstract  language  in  illus 
trating  the  nature  of  a  variable  quantity,  let  us  use  the 
following  problem :  — 

A  fisherman,  to  encourage  his  son,  promises  him  5  cents 
for  every  throw  of  the  net  by  which  he  shall  take  any  fish, 
but  the  son  is  to  remit  to  the  father  3  cents  for  each  unsuc 
cessful  throw  ;  after  12  throws  they  settle,  when  the  father 
pays  the  son  the  amount  of  the  agreement,  which  proved 
to  be  28  cents ;  what  was  the  number  of  the  successful 
throws  of  the  net  ? 

Let  x  =  the  number  of  successful  throws, 
then  bx  —  3  (12  —  a)  =28, 
that  is,  virtually,  8  a  —  36  =  28, 
which  is  an  equation  of  the  First  Degree. 

4t   Here  we  have,  derived  from  the  conditions   given, 


A   VARIABLE.  3 

an  expression  for  the  final  sum  paid  the  son,  which  expres 
sion  would  not  be  different,  whatever  that  sum  might  have 
been  found  in  the  event  to  be.  The  expression 

5z  — 3  (12  — aj), 

has  a  specific  value,  because  it  is  equated  with  28 ;  con 
sequently  x  is  found  to  have  the  specific  value  8.  If 
there  had  been  a  reserve  in  the  problem  whereby  the 
amount  of  the  payment  had  not  been  declared,  or  if  the 
formula  for  payment  were  to  remain  for  any  number  of 
trials  of  the  12  throws,  in  accordance  with  which,  28  cents 
could  hardly  be  expected  to  be  the  sum  paid  at  each  settle 
ment,  we  should  still  be  able  to  determine  all  the  relations 
between  the  changes  of  this  sum  due  the  son,  and  the 
changes  of  the  number  of  successful  throws. 

5.  If  the  amount  paid  or  to  be  paid,  were  to  be  styled 
a  sum  of  money,  or  y  cents,  we  might  look  upon  the  ex 
pression 

5x  —  3  (12  —  x)=y 

as  one  in  which  neither  x  nor  y  has  a  determinate  value; 
but  they  have  exactly  determinate  relative  values.  If  x 
or  the  number  of  successful  throws,  receive  original  sup 
positions  of  value,  y  or  the  formula  has  an  inferred  or  a 
relative  value : 

If  x  =  0,  then  y  =  —  36  cents. 
«'aj=  4£,  "  y—  0  " 
«  x=  14,  "  y=  76  " 
"  x  =  —  58,  "  y=—  500  " 
"  x=  12,  "  y=  60  " 
"  x=  11,  "  y=  52  « 

It  is  at  once  evident  that  if  x  be  increased  by  1,  y  will  be 
found  increased  8  times  as  much,  each  of  them  in  their 
respective  kinds  of  units. 

0.   There  is  no  algebraic  limit  to  this  relation  of  the 


4  DIFFERENTIAL   CALCULUS. 

change  in  y  being  8  times  the  change  in  &,  between  infi 
nite  positive  and  negative  values  of  either ;  nor  any  in 
terruption  in  the  case  of  fractional  values. 

The  disability  of  executing  in  the  problem,  fractional 
and  negative  values  in  x9  is  not  one  which  affects  in  the 
least  the  algebraic  expression,  or  the  truth  of  its  most  gen 
eral  indications.  For  when  the  successful  throws  are  as 
sumed  to  be  14  (although  the  whole  number  is  but  12),  so 
that  the  unsuccessful  ones  must  be  algebraically  expressed 
12  — 14  or  —  2,  in  accordance  with  which  view,  76  cents 
are  paid  the  son  at  the  settlement,  the  explanation  that  the 
indication  is  correct,  is :  for  every  unsuccessful  throw  the 
sum  paid  to  the  son  is  algebraically — 3  cents;  if  there  be 
—  2  of  such  throws,  then 

—  3X—  2  =  6; 
now  6  is  the  number  of  cents  by  which 

5  X  14,  as  found 
in  5  X  14  —  3  (12  —  14)  =  76 

is  properly  increased.  So  that  the  unlimited  amount  which 
may  be  indicated  as  payable  to  the  son  at  a  settlement, 
logically  agrees  with  a  correspondingly  unlimited  number 
of  successful  throws,  which  may  be  supposed,  as  indeed  it 
ought. 

7.  In  the  really  strict  use  of  common  language,  the  limits 
for  the  sum  paid  to  the  son  at  a  settlement,  are  from  0  to  60 
cents.  In  the  language  of  algebraic  equivalents,  there  are 
no  limits  whatever,  in  connection  with  the  supposed  prob 
lem,  because  its  indeterminate  quantities  so  far  enter  into  an 
equation  of  the  First  Degree. 

The  first  member  of  the  equation 

5«  —  3  (12  —  x)—y 
expressing  the  amount  payable  to  the  son  in  form  and 


A    VARIABLE.  5 

detail,  constructed  with  the  indeterminate  quantity  x,  in 
connection  with  other  determinate  quantities,  so  that  the 
expression  varies  when  that  quantity  x  may,  is  an  instance 
of  a  function  of  a  variable,  which  variable  x  may  be. 
Using  y  as  the  equivalent  of  the  function  in  amount  only, 
we  may  call  y  that  function,  when  thus  equated. 

8.  A  variable  is  a  quantity  which  may  have  different 
values,  and  is  represented  by  the  late  letters  of  the  alpha 
bet,  x,  by  y  or  by  z. 

9.  A  constant  quantity,  whether  known  or  unknown, 
is  one  which  by  original  assumption  is  not  to  vary  during 
an  investigation  into  which  it  enters,  or  by  inference  is  found 
to  be  so  conditioned  as  not  to  vary,  and  when  known,  it 
is  represented  by  number,  as  1,  or  20,  or  by  a  or  b,  etc. ;  d, 
however,  is  used  to  signify  differential,  and  its  use  purposely 
avoided  as  any  quantity  of  itself. 

10.  When  several  quantities  are  so  related  as  to  deter 
mine  one  another,  or  to  depend  on  one  another  by  equa 
tion  ;  a  variety  of  mutual  investigations  may  be  instituted 
among  them,  by  arbitrarily  assuming  an  independent  varia 
ble  or  variables,  and  then  examining  the  law  of  the  varia 
tion  of  the  dependent  variable. 

11.  Recurring  to  the  problem  of  the  fisherman,  we  have 
the  equation 

5;c  — 3  (12  —  x)=  28, 

which  answers  its  algebraic  purpose  of  determining  one 
specific  value  for  x,  which  is  8,  the  number  of  successful 
throws  of  the  net.  If  8  be  supplied  in  the  place  of  x,  and 
any  one  of  the  other  quantities  of  the  equation  were  left 
unstated  in  the  conditions,  and  be  made  a;  in  a  new  investi 
gation,  it  could  be  determined.  Indeed  if  any  two  of  the 
quantities  were  unstated  in  amount,  the  mutual  dependency 
of  the  two  becomes  evident.  For  a  conventional  reason  we 


6  DIFFERENTIAL  CALCULUS. 

will  call  that  quantity  cc,  to  which  we  may  wish  to  reserve 
the  right  to  assign  arbitrary  values,  to  the  extent  that  we 
can  do  so,  and  will  call  the  other,  which  receives  inferred 
values,  y.  The  exchange  of  x  for  y  we  will  call  the  con 
verse  of  the  investigation.  When  yean  be  isolated  as  one 
member  of  the  equation,  not  occurring  in  the  other,  the 
function  of  x  is  called  explicit.  We  will  suppose  the 
explicit  form  of  the  functions  of  one  variable  aj,  implied  in 
some  of  the  following  statements  of  the  above  equation,  to 
be  worked  out,  and  we  will  suppose  their  significance  to  be 
enunciated  in  words. 

1.  5£c  —  3  (12  —  x)=   y. 

2.  x.  8—     (12  —  8)=   y. 

3.  5X8  —  3    (oj  — 8)=   y. 

4.  5X  8  — a  (12  —  8)=   y. 

5.  $y  —  x  (12  —  8)  =  28. 

6.  8y  —  3    (x  —  8)  =  28. 

7.  5cc  —  y  (12  —  x)  =  2S. 

8.  5X8  —  y    (oJ  —  8)  =  28. 

9.  by  —  3    (x  —  y)  =  28. 
10.                       x  .  y  —  3  (12—  y)  =  28. 

Each  of  the  above  has  evidently  its  converse. 

It  may  be  useful  to  adopt  a  comprehensive  enunciation 
of  the  problem. 

In  the  parentheses  (  )  which  follow  in  the  problem,  let 
the  following  reading  be  understood :  the  variables  now 
being  supposed  to  be  two,  one  depending  on  the  other,  viz : 

(For  such  number  is  exactly  compatible  with  the  other 
numerical  quantities,  received  as  given,  without  regard  to 
the  brackets.) 


A   VARIABLE.  t 

In  the  brackets  [  ]  let  the  following  reading  be  under 
stood  : 

[Or  an  indefinite  numerical  quantity,  if  in  one  of  the 
other  brackets,  another  indefinite  numerical  quantity  be 
understood  to  be  read.] 

12.  A   fisherman  promises  his    son    5    cents  (     )   [     ] 
for  every  throw  of  the  net  by  which  he  shall  take  any 
fish,  but  the  son  is  to  remit  to  the  father  3  cents   (     ) 
[     ]  for  every  unsuccessful  throw ;  after  12  throws  (     ) 
[     ]  they  settle,  when  the  father  pays  the  son  the  amount 
of  the  agreement,  which  was  28  cents  (     )   [      ],  there 
proving  to  have  been  8  (     )  [     ]  successful  throws.     Re 
quired  all  the  truths  dependent  on  the  various  readings, 
for  each  two  of  these  indeterminate  but  mutually  depen 
dent  quantities. 

All  of  the  above  equations  and  their  converses,  except 
the  8th  and  10th  are  of  the  First  Degree,  and  the  variables 
in  them  may  have  any  algebraic  values,  positive,  negative, 
or  infinite,  and  they  vary  at  uniform  rates,  in  passing  from 
one  value  to  a  succeeding  one. 

In  the  8th  and  10th  and  their  converses,  since  the  varia 
bles  are  factors  together,  the  equations  are  of  the  Second 
Degree,  and  their  variations  are  subject  to  other  laws. 

13.  Such  relative  rates  of  variation  are  subject  to  exact 
numerical  determination,  which  it  is  the  object  of  the  suc 
ceeding  Sections  to  explain  in  their  general  nature. 

14.  It  is  now  evident  that  those  problems  in  algebra  which 
•  are  based  upon  Simple  Equations  or  those  of  the  First 

Degree,  are  such  only  with  reference  to  the  one  investiga 
tion  for  which  they  are  offered,  and  that  a  change  of  the 
investigation  for  determining  other  relations  between  quan 
tities  fully  conditioned  in  such  problem,  is  likely  to  require 
the  use  of  Aifected  Equations,  which  are  always  as  high 
as  the  Second  Degree,  and  may  be  higher. 


8  DIFFERENTIAL  CALCULUS. 


SECTION  II. 

DEFINITIONS    RELATING    TO    FUNCTIONS.  —  THE    USE 
OF   SIGNS. 

15.  An  explicit  algebraic  function  of  a  variable  quantity, 
is  an  indicated  mode  in  which  addition,  subtraction,  multi 
plication,  division,  and  other  arithmetical  processes  with 
quantities,  either,  any  or  all  of  them,  in  or  among  which, 
said  variable  is  somehow  concerned,  are  used  for  working 
out  a  resultant  quantity,  and  among  which  indicated  opera 
tions,  this  variable  quantity  holds  a  marked  position,  as  the 
one  to  which  a  particular  reference  is  to  be  made  in  regard 
to  its  changes  of  value. 

16.  Hence  a  function  is  primarily  and  always  a  mode 
of  constituting  a  quantity  ;  although  a  function  of  a  variable 
may,  after  some  hypothesis  for  itself  or  that  variable,  pos 
sess  a  value,  the  function  cannot  be  considered  as  merged 
in  a  specific  amount  of  quantity.     We  cannot  speak  of  a 
great  or  of  a  small  function,  as  some  writers  do  because  that 
means  a  great  mode^  —  a  mode  is  not  a  value  or  a  quantity. 

17.  Strictly  speaking,  a  function  of  a  variable,  having  as 
it  may,  constant  quantities  in  it,  is  a  function  of  all  the  dis 
tinct  quantities  in  it,  because  it  requires  them  all  to  per 
form  the  office  of  representing  some  quantity  in  mode,  but 
custom  has  sanctioned  calling  the  whole  expression,  that 
determines  some  designed  quantity,  in  which  the  variable 
occurs,  a  function  of  the  variable,  although  the  variable 
occurs  in  no  more  than  in  one  of  the  terms  of  such  ex 
pression,  and  in  the  simplest  manner;  and  however  com 
plicated  other  terms  containing  constants  may  be. 

18.  Functions  which  are  not  algebraic  have  restrictions 
to  Geometry,  Trigonometry,  or  to  Logarithms,   which    are 
to  be  considered  in  this  treatise.     Functions  which  are  not 


DEFINITIONS  RELATING  TO   FUNCTIONS.  9 

explicit,  may  be  implicit,  or  involved  in  an  equation,  and 
determinate  as  a  mode,  only  after  algebraic  process.  The 
implicit  is  the  most  general  and  comprehensive.  It  is  often 
subject  to  difficulty  of  solution,  or  to  impossibility. 

19.  A  formula  is  an  expressed  mode  of  operating  with 
quantities,  for  deducing  the  amount  of  another.     In  a  for 
mula,  as  such,  there  is  not  necessarily  a  quantity,  to  which 
any  reference  is  to  be  made  as  subject  to  change  of  value. 

An  explicit  function  is  a  formula,  in  which  there  is  a 
quantity,  subject  to  a  change  of  value,  and  subjecting  the 
formula  to  a  change  of  value. 

20.  There  is  no  absolute  need  of  any  equation  in  the 
statement  of  an  explicit  function.     The  equating  of  it  with 
its  own  correspondingly  variable  amount  called  by  another 
name  y,  is  often  only  a  piece  of  convenience.     Whenever  a 
function  of  a  variable  appears  equated  with  a  determinate  or 
constant  quantity,  supposition  is  evidently  applied  for  the 
value,  or  values  if  there  should  be  more  than  one,  of  that 
function.     If  that  constant  is  not  removable,  or  open  to 
supposition  for  a  change  of  value,  the  variable  x  may  take 
the  name  of  unknown  quantity ;  although,  since  it  is  deter 
minate  it  might  be  regarded  as  known,  in  the  calculus.    We 
sometimes  use  a  function  of  a  constant ;  this  is  when  such 
constant  occupies  only  temporarily  the  place  of  a  well  un 
derstood  variable. 

21.  When  the  conditions  of  an  algebraic  problem  are 
stated  in  the  form  of  an  equation,  the  members  of  such 
equation  may  each  be  functions  of  the  unknown  quantity, 
but  may  be  reduced  to  one  function,  for  the  value  of  some 
other  quantity. 

22.  A  marketman  purchased  fowls :  some  2  for  a  dollar, 
and  as  many  more  3  for  a  dollar,  and  sold  them  at  the 
rate  of  5  for  2  dollars,  losing  4  dollars  by  the  operations ; 
required  the  number  of  each  sort. 


10  DIFFERENTIAL   CALCULUS. 

Let  x  =  the  number  of  each  sort; 

then  we  have  these  two  expressions  for  one  sum  of  money, 
which  may  be  put  equal,  viz : 


In  case  the  number  of  each  sort  or  x  be  variable,  these 
two  expressions  will  permit  the  forming  of  one  function  of 
JG,  for  the  sum  lost,  viz  : 


This  function  remains  the  same  if  the  sum  lost  were  in 
definite,  and  were  to  change  only  on  change  of  the  num 
ber  of  each  sort  ;  in  the  supposition  it  is,  when  without  an 
equation  : 

X       i         X  4:X 

~2    '      3          5 

and  it  is  ready  for  the  comparison  of  the  number  of  dollars 
lost,  during  changes  of  the  number  of  each  sort.  It  will  be 
convenient  to  equate  it  with  y,  as  being  also  the  number 
of  dollars  lost. 

33,  The  problem  can  be  enunciated,  in  the  general  aspect, 
for  algebraic  determinations,  and  for  the  calculus  deter 
minations,  as  before,  with  the  fisherman  problem. 


34.  All  strictly  algebraic  quantities  in  the  Differential 
Calculus,  are  subject  to  algebraic  expression,  and  are  numer 
ical  in  their  nature,  and  are  real  or  imaginary  or  irrational 
in  value.  All  functions  have  numerical  amounts,  or  ex 
pressions,  for  their  values,  —  when  rendered  determinate 
and  real. 

25.  Three  triangularly  placed  points  (/.)  are  used  to  sig 
nify  hence  or  therefore. 


THE  USE  OP  SIGNS.  11 

26.  One  of  these  characters,  >  or  <,  is  placed  between 
two  unequal    quantities,  the    larger  quantity  of  the  two 
being  embraced  by  the  limbs  of  the  character,  and  may 
be  ^read,  —  being  greater  than,  —  being  less  than,  —  or,  is 
greater  than  or  less  than. 

27.  Zero,  or  naught,  is  freely  treated  as  a  determinate 
value  for  quantity.     Hence,  in  the  expression  0  >  a,  a  is 
negative.     A  quantity  that  is  negative  is  freely  called  less 
than  nothing,  for  the  uniformity  and  brevity  of  the  mode 
of  expression.     When  zero  forms  one  member  of  an  equa 
tion,  it  both  marks  and  simplifies  it,  for  some  general  uses  ; 
hence  its  use. 

28.  A  single  point    ( . )  is  used  as  an  abridgment  of  the 
sign  of  multiplication  X-    When  the  point  is  placed  directly 
between  figures,  it  ought  to  indicate  decimals,  with  never 
theless,  easily  understood  exceptions. 

29.  A  prostrate  figure  of  co  signifies  an  infinite  quantity. 

30.  The  sign  of  equality,  =  may  often  be  advantageously 
read  as  a  verb  equals,  sometimes  as  equalling,  or  as,  that  is, 
or  as,  that  is  to  say. 

31.  The  Binomial  Theorem  should  be  mentioned  as  the 
Foundation  of  the  essential  principles  of  the  Calculus,  and 
is  demonstrated  in  most  treatises  of  algebra,  as  related  to 
indexes  being  whole  and  positive  numbers. 

The  extension  of  its  demonstration,  to  embracing  bino 
mials  having  fractional,  negative,  or  imaginary  indexes  is 
commonly  made  in  the  Calculus  as  a  sequence  to  Mac- 
laurin's  Theorem. 

The  following  ocular  views  of  one -application  of  the 
Binomial  Theorem,  will  impress  its  law  more  concisely 
than  the  use  of  n,  n—  1,  n  —  2,  etc.,  will  do. 


*T   * 


12  DIFFERENTIAL   CALCULUS 

32.  The  differences  in  the  successive  supposed  values  of 
a  quantity,  may  obviously  be  fractionally  minute ;  the 
conceiving  of  a  quantity  at  such  successive  values,  gives  rise 
to  the  idea  and  expression,  of  the  growth,  decrease,  etc.,  of 
the  quantity.  It  was  this  unnecessary  transfer  of  the  mode 
of  apprehending  quantity  in  this  state  by  the  mind,  to  the 
quantity  itself,  that  gave  rise  to  considering  such  quantities 
as  "generated  by  motion,"  —  "  the  quantity  thus  generated 
is  called  the  fluent  or  flowing  quantity,"  —  "  the  velocities 
with  which  flowing  quantities  increase  or  decrease  at  any 
point  of  time,  are  called  the  fluxions  of  those  quantities 
at  that  instant."  (Vince's  Fluxions.)  These  forms  of  ex 
pression  have  been  wisely  discontinued ;  although  a  modified 
form  of  these  expressions  is  sometimes  convenient,  such  as 
growth  — faster  —  and  slower,  of  the  value  of  a  function, 
or  of  a  quantity. 


SECTION   III. 

THE    DEGREES    OF    AN    EQUATION.  —  FIRST  AND    SEC 
OND    DEGREES. 

33.  An  algebraic  equation  of  the  First  Degree  contain 
ing  one  quantity  called  unknown  (but  which,  however,  may 
be  entirely  determinate,  having  a  fixed  value)  in  the  form 
best  adapted  to  exhibit  its  degree,  and  consequently  to  offer 
the  best  opportunity  for  showing  the  nature  of  its  quanti 
ties,  is  by  general  expression 


in  which  A  embraces  the  algebraic  aggregate  of  all  given 
quantities,  which  are  factor  with  x,  and  B  represents  gen 
erally  all  other  quantities,  and  each  term  has  the  sign  + 


FIRST   AND   SECOND   DEGREES.  13 

(plus)  in  the  sense  that  is  understood  to  embrace  minus,  in 
case  the  particular  conditions  require  it.  It  is  evident  that 
the  value  of  x  is  determinate  when  A  and  B  are  not  each 
at  once  zero  or  infinite. 

PROBLEMS. 

34,  It  is  required  to  determine  whether  the  following 
equations  are  of  the  First  Degree,  and  in  what  form  A.  and 
B  are  represented. 

O+VT)  . 
1.  c  -       —  -f-  c  =  150  a. 


35.   The  general  equation  of  the  First  Degree  contain 
ing  two  quantities,  x  and  y,  indeterminate  in  value  is, 

in  which  A  is  the  aggregate  of  given  quantities,  factor  to 
a;,  and  B  of  given  terms,  factor  to  y,  and  in  which  C  is  the 
aggregate  of  terms  given  separately.  Only  the  quantities 
implied  by  A,  B,  and  C  being  known,  x  and  y  cannot  be 
determined  by  them.  Nevertheless  it  is  evident  that  there 
is  a  relation  always  implied  between  the  values  of  x  and  y, 
which  the  equation  and  the  fixed  values  of  A,  B,  and  C 
preserve. 

30,  In  the  particular  case  in  which  A  or  B  should  have 
the  value  zero,  x  or  y  receives  a  fixed  value.  It  is  in 
tended  that  A)  B,  and  C  should  represent  determinate  and 
unchangeable  quantities  in  any  particular  use,  and  should 
be  general  only  for  the  expression  of  a  general  formula.  In 
the  particular  cases  in  which  A  should  be  infinite,  while  B 
and  C  are  not,  x  loses  general  values,  and  must  become 
zero.  In  the  particular  case  in  which  B  should  be  infinite, 
while  A  and  C  are  not,  y  loses  general  values,  and  must  be 
come  zero.  In  the  particular  ease  when  C  is  infinite,  while 
2 


14  DIFFERENTIAL   CALCULUS. 

A.  and  J5  are  not,  both  x  and  y  may  be  infinite,  and  one  of 
them  must  be  infinite. 

37.  In  the  particular  case  when  A  and  C  are  each  infi 
nite,  while  IB  is,  or  is  not  infinite,  since  x  becomes 

By       c 

*c/  —  —  —  . 

A         A' 

it  will  be  indeterminate. 

The  quantities  of  which  A,  It,  and  C  are  the  represen 
tatives,  if  definite  in  value  may  be  called  constant  with 
reference  to  any  investigation,  for  which  a  purpose  is  sub 
served  in  making  them  so,  while  x  and  y  may  vary  in  value, 
but  evidently  with  a  dependence  upon  each  other ;  they 
may  be  variables. 

38.  THEOREM.     In  an  equation  of  the  First  Degree  be 
tween  two  variables,  if  one  variable  be  supposed  to  change 
uniformly  in  value  from  any  supposed  definite  value,  the 
other  must  change  uniformly  in  value. 

39.  The  variables  in  an  equation  of  the  First  Degree  may 
have  the  utmost  range  of  values. 

40.  But  in  reference  to  the  amount  of  variation,  that  va 
riable  x,  which  is  factor  to  A,  will  vary  uniformly  as  many 
times  y  (the  factor  of  J5),  as  is  expressed  by  the  quotient 

TJ 

-,  and  they  need  not  concur  in  increase  or  decrease,  as  the 

•A. 

mode  of  simultaneous  variation. 

41.  In  the  particular  case  when  A  =  l  and  _Z?  —  1,  and 
y  is  negative,  we  have  the  principle  of  Simple  Addition  in 
Arithmetic,  or 

x+C=y, 

when  it  is  evident  that  if  x  be  increased,  y  is  dependency 
increased.  Because,  if  one  of  two  numbers  which  are  pro 
posed  to  be  added,  be  first  increased,  the  amount  is  as 
much  increased.  If  x  be  negative,  the  equation  of  the  First 


THE  DEGREES  OP  AN  EQUATION.          15 

Degree  as  above,  becomes  the  representative  of  the  nature 
of  Simple  Subtraction  in  Arithmetic,  with  a  properly  cor 
responding  result. 

42.   In  the  particular  case  when  A  or  B  is  negative  and 

(7=0,  we  have 

Ax 


A  x 

—  being  like  two  factors  in  Simple  Multiplication,  and 

y  their  product,   and  it  is  evident   that  if  x  change   in 

value,  y  or  the  product  will  change  —  times  as  much  as 

B      A 

the  one  factor  x.     But  it  is  evident  that  —  may  be   less 

B 

than  a  unit. 

43.   The  general  equation  of  the  Second  Degree  between 
two  variables  x  and  y  is 

=  0, 

where  in  the  particular  case  of  A  =  0,  C=Q,J3  =  —  1, 
D  =  0,  and  E=  0,  we  have 


which  is  the  representative  of  the  nature  of  Simple  Divis 
ion  in  Arithmetic,  if  we  propose  to  let  a  divisor  and  the 
quotient  vary,  and  the  dividend  remain  unchanged. 

44,  Here  we  may  expect  some  law  of  mutual  variation 
between  x  and  y,  different  from  what  appears  in  respect  to 
them,  in  equations  of  other  degrees. 

If  we  take  any  number,  as  257,  as  dividend,  and  divide 
it  by  any  number,  as  18,  and  then  let  the  divisor  be  changed 
by  a  specific  amount,  3,  we  have 

257-^18  = 
257  -7-21  = 


16  DIFFERENTIAL   CALCULUS. 

and  again,  divide  it  by  any  number,  as  8,  and  by  8  +  3, 
we  have  . 

257  -^    8  =  32£, 

257  -r  11  =  23  A, 

where  we  soon  perceive  that  the  change  in  the  quotient  is 
not  uniform  ;  and  we  should  find  that  according  as  the  di 
visor  and  quotient  approach  equality  with  each  other,  will 
the  rate  with  which  the  larger  decreases,  diminish  with  a 
retarding  rate,  compared  with  the  less,  supposed  to  vary 
with  an  assumed  uniformity,  which  is  both  a  possible  and 
necessary  mode  of  making  the  supposition. 

45.   In  the  particular  case  where  A  =  Q,  B  =  0,  (7=1, 
D  =  —  1,  and  F=  0,  we  have 


.  e.,  #=2A 

in  the  examination  of  which  we  shall  find  that  the  second 
or  square  roots  of  a  quantity,  do  not  vary  uniformly  with 
the  quantity. 

46.  The  complete  discussion  of  the  equation  of  the  Sec 
ond  Degree,  is  comprehensive  of  a  protracted  variety  of 
particular  cases  and  principles. 

In  the  use  that  is  to  be  made  of  'functions  of  variables^  we 
may  not  be  able  to  know  all  of  them  by  classification  as 
of  any  numerical  degree  that  has  been  numbered  or  defined, 
although  quite  desirable  when  it  may  be  readily  determined. 

47,  When  the  variables  x  and  y  in  an  equation  are  at 
powers  expressed  by  whole  and  positive  indexes,  the  nu 
merical  name  of  its  Degree  is  determined  by  the  highest 
sum  of  the  indexes  of  the  variable  or  variables  which  are 
direct  factors  in  any  one  term,  with  A  or  .#,  etc. 


FIRST   AND   SECOND   DEGREES.  17 

PROBLEMS. 

48.   It  is  required  to  determine  of  what  Degree  are  the 
equations : 


2.  *abx          *  =     *       a. 


ax  —  2         (a  +  b)xy 

The  general  equation  of  the  Second  Degree 
(1.)     Ay*  +  3xy+Cx*  +  Dy  +  E 
when  y  is  equated  with  its  value,  gives  us 

(2.)    y  =  _£f±5±  -Lv   j  (^  - 

/  ^1  /  yi  (^ 

2  (^  Z>  —  2  .4  j£)  z  +  D  2  —  4  A  F  I  . 


Now  the  quantities  represented  by  the  capital  letters, 
are  general  in  these  equations,  but  become  particular,  in  a 
given  applied  case.  According  to  the  relative  value  of 
these  constants,  a  classification  of  the  nature  of  the  Sys 
tems  of  values  of  y  and  of  x  may  be  made.  Of  these, 
three  systems  of  values  for  y  and  x  are  prominent,  condi 
tioned  thus  :  they  are, 

1.  when  B*  —  4^L<7<0,  Elliptical  ; 

2.  «      B  2  —  4  A  C">  0,  Hyperbolic  ; 

3.  «      B  2  —  4  A  O=  0,  Parabolic. 

40.    Since  the  general  equation  is  symmetrical  with  refer 
ence  to  y  and  x,  these  variables  have  corresponding  sys 
tems  of  values,  which  values  are  equal  when  the  constants 
2* 


18  DIFFERENTIAL   CALCULUS. 

A  =  C  and  D  —  E,  and  retain  a  similar  nature  when  these 
constant^  are  not  equal. 

50.  In  the  elliptical  condition,  both  x  and  y  have  a  maxi 
mum  and  a  minimum  value. 

51.  In  the  hyperbolic,  there  are  two  systems  of  symmet 
rical  values  for  each  variable,  isolated  from  each  other; 
and  if  there  be  a  maximum  for  one  of  them  there  must  be 
a  minimum  also  for  it,  and  there  may  or  may  not  at  once 
be  a  maximum  and  minimum  also  for  the  other. 

52.  In  the  parabolic,  there  must  be  a  maximum  or  mini 
mum  for  one  of  the  variables,  and  there  may  be  for  the 
other. 

53.  In  the  general  equation  there  may  be  only  imaginary 
values  to  one  of  the  variables,  when  there  must  also  be  to 
the  other ;  these  occur  when  the  quantity  under  the  radical 
sign  is  found  negative  in  equation  (2.) 

In  geometry  much  importance  is  attached  to  the  indica 
tions  of  these  equations,  in  the  study  of  Curves  of  the 
Second  Degree.  The  terms,  maximum  and  minimum, 
will  be  seasonably  defined. 


SECTION   IV. 

INCREASE  OF  FUNCTIONS.  —  DECREASE  OF  FUNCTIONS. 
-STATIONARY  VALUES   OF  THEM. 

EXERCISES. 

54.  1.  There  are  purchased  99  pounds  of  a  commodity 
at  a  price  fixed  to-day,  but  liable  at  the  next  purchase  to  be 
found  changed.  How  many  times  more  will  the  value  of 
the  99  pounds  vary,  than  the  value  of  a  single  pound  ? 


INCREASE   OF   FUNCTIONS.  19 

2.  How  many  times  more  will  the  value  of  17,000 
pounds  change  than  the  value  of  1  pound  of  it  ? 

3.  How  will  the  value  of  one  sixteenth  of  a  pound  vary 
compared  with  the  value  of  one  pound  of  it  ? 

4.  How  will  the  capacity  of  100  equal  casks  vary  com 
pared  with  the  capacity  of  one,  if  the  casks  should  be  made 
larger,  on  a  repetition  of  the  manufacture  ? 

5.  How  will  b  x  change  if  x  change  ? 

6.  How  will  (a  -f-  b  —  c)  x  vary  in  amount  if  x  grow 
greater  ? 

7.  If  A  expresses  the  amount  that  x,  in  the  above  twro 
expressions    containing   #,    changes,    what    expresses   the 
amount  of  the  expression's  change  ? 

8.  When  a  commodity  is  worth  10  cents  a  pound,  and 
it  is  quoted  as  tending  upward  in  price,  does  the  value  of 
100  pounds  commence  to  increase  any  faster  or  slower  than 
it  would  if  the  price  were  15  cents  per  pound,  and  the 
value  of  the  100  pounds  commenced  to  increase  from  that? 
That  is,  does  the  ratio  of  the  growth  of  100  x  to  that  of 
1  x  depend  on  the  greatness  of  the  quantity  jc,  or  on  the 
amount  of  the  growth  of  x  ? 

9.  Does  that  function  of  cc,  which  100  x  is,  grow  uni 
formly,  on  uniform  increase  of  x  ? 

10.  Will  8  x2  grow  uniformly  on  growth  of  x  ? 

11.  How  great  could  the  value  of  1542  pounds  of  a  com 
modity  ever  become,  by  an  indefinite  increase  of  the  price 
of  one  pound  ? 

12.  If  x  varies,  which  of  these  two  functions  of  aj,  viz., 
10  x  and  11  £c,  will  vary  the  more  in  amount  ?     How  much 
more  ? 

13.  If  the  value  of  71  pounds  of  a  commodity  be  nega 
tive,  or  be  considered  subtractive  from  some  greater  sum, 
would  an  increase  of  the  value  of  the  71  pounds  be  nega 
tive  ? 

14.  How  would  the  amount  of  the  decrease  of  the  value 


20  DIFFERENTIAL   CALCULUS. 

of  the  71  pounds  stand,  as  affecting  the  original  greater 
quantity  ? 

55.  Both  the  number  of  pounds  of  a  commodity  and  the 
number  of  cents  per  pound,  will  now  be  conditioned  to  in 
crease  ;  if  there  are  u  pounds,  at  x  cents  per  pound,  and 
the  pounds  increase  by  any  amount  &,  and  the  cents  per 
pound  increase  by  any  amount  A,  what  expresses  the  new 
entire  value  ?  What  expresses  simply  the  increase  of  value  ? 

1.  If  the  amounts  k  and  h  be  considered  alike,  or  each 
h,  how  does  the  expression  of  the  increase  of  the  value  of 
the  commodity  become  united  ? 

Ans. From  ux-\-uh-\- x k-\-h k  to  u x-\- (u-\-x) h-\-h*. 

2.  How  does  the  expression  still  more  condense,  if  the 
original  number  of  pounds,  and  number  of  cents  per  pound, 
were  alike,  or  each  x ?  Ans.    Into  a?2-[-2ajA-|-A2. 

3.  What  are  the  factors  of  just  the  increase  ? 

Ans.  h  and  2  x  -\-  h. 

56.  Thus  far  the  quantities  u  and  k  x  and  h  may  be  any 
whatever.     But  a  quantity  situated  and  treated  like  h  as  an 
increment  of  a*,  may  be  moulded  to  a  purpose,  —  a  merely 
mathematical  purpose,  —  be  a  creature  of  supposition,  sub 
servient  to  the  illustration  of  a  mere  addition  to  a?,  and  may 
be  indefinitely  small ;  while  x  is  a  quantity  to  be  accepted 
as  given  ;  it  is  any  quantity,  till  made  determinate  by  some 
condition. 

57.  If  there  are  x  pounds  of  a  commodity  at  x  cents  per 
pound,  and  x  be  increased  an  indefinitely  small  amount  A; 
by  how  many  times  as  much  as  A,  at  the  least,  will  the  value 
of  the  whole  of  this  commodity,  commence  to  increase  ? 

Ans.  2  x. 
How  many,  to  decrease,  if  x  diminish  as  much  ? 

Ans.  2#. 

1.   If  the  number  of  pounds  be  restored  as  w,  while  k 


DECREASE   OF   FUNCTIONS.  21 

may  be  h  according  to  the  above  indication,  how  many 
times  as  much  as  7i,  at  least,  will  the  value  of  the  whole 
commodity  commence  to  increase  ?  Ans.  x  -\-  u. 

2.  How  many,  if  u  increase  while  x  diminishes  ? 

Ans.  —  x  -f-  u. 

3.  How  many,  if  u  decrease  and  x  increase  ? 

Ans.  x  —  u. 

4.  If  x  pounds  of  a  commodity,  worth  x  cents  per  pound, 
are  possessed  by  each  of  9  persons,  how  will  the  whole 
value  commence  to  decrease,  if  x  decrease  by  (an  amount 
that  is  merely  nominal)  h  ?  Ans.  9  X  2  x  times  h. 

5.  If  one  of  these  persons  obtains  his  share  from  a  lot 
of  20  pounds  of  the  commodity,  all  of  the  above  value,  how 
many  pounds  does  he  take  when  he  leaves  the  greatest 
possible  value  possessed  by  that  remaining  lot  ? 

6.  In  20  x  —  x 2  apply  successively  1,  2,  3,  etc.,  for  cc, 
till  the  greatest  remainder  be  found. 

7.  If  a  piece  of  ground  conditioned  to  be  kept  square, 
is  to  be  enlarged,  on  how  few  of  its  sides  must  the  new 
area  commence  to  form  ?     On  how  few,  if  the  square  must 
decrease,  must  the  deductive  area  commence  to  form  ? 

8.  There  is  a  rectangular  piece  of  ground  10  rods  in  one 
dimension ;  its  other  dimension  is  the  same  and  to  be  kept 
the  same,  as  the  side  of  a  certain  square  piece  of  ground ; 
upon  the  suggestion  that  the  sides,  one  of  each  lot  which 
are  alike,  commence  to  increase,  which  lot  commences  to 
increase  the  faster  in  its  area  when  that  common  side  is  4 
rods  ?    When  6  rods  ?    When  2  rods  ?    When  1000  rods  ? 

9.  What  number  of  rods  must  that  common  side  con 
tain,  when  the  greater  rapidity  of  the  growths  of  the  two 
lots,  passes   from  one  to   the    other?     From  which,    to 
which  ? 

10.  When  the  side  of  the  square  lot  is  50  rods,  and  the 
lot  commences  to   increase,  will   the  product  expressing 
the  increase  of  area,  have  one  of  its  factors  greater  than 


22  DIFFERENTIAL   CALCULUS. 

if  the  side  was  49  rods,  and  the  increase  then  took 
place  ? 

11.  If  we  have  this  function  of  a;,  viz.,  92  x  —  x  2,  con 
sisting  of  a  minuendive  and  a  subtrahendive  term,  must 
the  value  of  the  function,  which  evidently  consists  of  their 
difference,  always   increase  while   the    minuendive   term 
increases  the  faster  of  the  two  ? 

12.  Must   its   value  decrease  when   the  subtrahendive 
term  increases  the  faster  ?     Can  a  value  of  x  and  of  the 
function  be  found,  when  the  increase  of  the  subtrahendive 
term,  being  negative,  is  equal  to  the  increase  of  the  minu 
endive  term,  which  is  positive,  so  the  sum  of  the  increases 
equals  0  ? 

PROBLEMS. 

58,  1.  Let  it  be  required  to  divide  the  number  92 
into  two  such  parts,  as  Avhen  multiplied  together,  shall 
produce  the  greatest  product. 

Let  x  =  one  of  the  parts. 
.-.  92  —  x  =  the  other. 
/.  92  x  —  x  2  =  their  product. 

When  the  function  is  the  greatest  in  value,  it  is  evident 
that  either  adding  or  subtracting  any  small  and  indefinite 
amount  A,  to  or  from  x,  diminishes  the  value,  so  that 

92  (x  +  h)  —  (aja  +  2  x  h  +  A2)  <  92  x  —  x\ 
and     92  (x  —  h)  —  (x*  —  2  xh  +  A2)  <  92  x  —  x*  ; 

therefore  in  accordance  with  what  has  been  said, 


and  92A>2xA  —  A2, 

where   all   the   signs   of  the   last   expression   have   been 


STATIONARY   VALUES.  23 

changed,  because  the  members  of  the  inequation  are  com 
pared  only  with  each  other.     Hence, 

92  <  2  x  4-  h, 
and  92>2aj  — A; 

hence,  x  =  46  within  one  half  of  the  smallest  quantity 
that  can  be  conceived  of. 

2.  A  boy  having  16  equal  parcels  of  marbles,  gave  to  as 
many  boys  as  there   were  marbles  in  a  parcel,  as  many 
marbles  each  as  there  were  boys,  and  retained  the  great 
est  possible  number  himself,  for  any  number  in  a  parcel. 
Required  the  whole  number  of  marbles ;  of  parcels ;  of 
boys;  number  of  marbles  given   away,  and   the  number 
retained.  Ans.  128,  8,  8,  64,  and  64. 

3.  Some  benevolent  persons,  gave  each  240  dollars  to 
some  orphans ;  four  times  as  many  orphans  each  month 
received  2  dollars  each,  and  during  three  times  as  many 
months  as  there  were  contributors ;  the  unexpended  sum 
was    still    the   greatest    possible.      How    many   persons 
gave?  Ans.   5. 

59.  It  is  generally  possible,  when  the  terms  of  a  func 
tion  are  simple  and  few,  not  fractional,  when  the  indices 
are  positive  and  entire,  to  foretell,  on  logical  principles, 
whether  it  is  a  maximum  or  minimum  that  a  function  has, 
if  but  one.  For  instance,  by  noting  the  sign  of  the  term 
having  the  variable  at  the  highest  positive  power,  we  shall 
perceive  that  at  some  positive  value  of  the  variable,  such 
term  will  rule ;  if  this  be  positive  the  presumptions  are,  a 
minimum  at  some  value  of  the  variable,  if  there  be  either, 
—  if  this  be  negative  the  presumption  is  reversed ;  such 
term  being  destined  to  infinity  in  itself,  wTill  leave  the 
maximum  or  minimum  behind. 

4.  Consider  x  2  —  92  x  in  these  regards. 


24  DIFFERENTIAL   CALCULUS. 

Infinite  and  positive  are  easily  inferred  ;  a  minimum  prob 
able.  If  the  function  be  considered  settled  at  a  minimum, 
then  x  takes  a  definite  value ;  and  if  x  be  increased,  the 
positive  member  must  increase  more  than  the  other 
diminishes  ;  so  that,  as  before, 

2x!i  +  hz>  92  A; 

and  again  retrospectively,  subtracting  the  last  that  x  gained 
in  its  growth,  for  the  function  by  hypothesis  did  diminish, 
and  the  negative  term  had  diminished  more  than  the  pos 
itive  increased,  so  that  we  have,  as  before, 

92  h>2xh  —  A  2, 
and  x  =  46  as  before. 

So  it  appears  that  nothing  is  different  in  discovering  a 
minimum,  from  discovering  a  maximum. 

The  consideration  of  a  function,  when  not  readily  dis 
closing  its  tendencies  in  these  respects,  especially  when  it 
probably  has  both  maxima  and  minima,  is  deferred  to 
another  section  of  the  treatise. 

5.  From  a  cistern  holding  6218  times  a  certain  measure 
full,  were  taken  away  that  measure  full  11  times  each  day, 
during  as  many  days  as  that  measure  held  gallons.     The 
greatest  possible  number  of  gallons  were  left  in  the  cistern 
at  last ;  then  how  many  quarts  did  that  measure  hold  ? 

Ans.    1130  T6T. 

6.  A  grain  dealer  sells  to  A  the  same  number  of  bush 
els  of  grain  out  of  100  bushels,  as  he  sells  the  remainder 
to  B  for,  in  cents  additional  to  25  cents  per  bushel,  and 
realizes  the  greatest  possible  sum  from  that  sold  to  B. 
Required  the  number  of  bushels  sold  to  A,  and  the  price 
per  bushel  of  that  sold  to  B. 

Ans.   37J  bushels,  and  G2£  cents. 

7.  From  a  cask  containing  19  times  a  certain  measure 


GREATEST   VALUES   OF    FUNCTIONS.  25 

full  of  water,  a  smaller  measure  by  7  quarts  is  drawn  full,  as 
many  times  as  it  held  quarts,  when  the  greatest  possible 
quantity  of  water  remained  behind  in  the  cask.  Required 
the  capacities  of  the  measures.  Ans.  16^-  and  9£  qts. 

8.  From  a  lot  of  1501  (a)  bushels  of  hay-seed  79  (b) 
casks  full  were  put  up  for  exportation ;  the  remainder  were 
sold  for  home   consumption,  at  3  (c)  dollars  per  bushel 
more  than  there  were  bushels  in  a  cask.     Required  to  ad 
just  all  the    quantities   which   are   not   stated   definitely, 
to  the  greatest  value  of  the  home  sale. 

Let  x  =  number  of  bushels  per  cask. 

/.  x  -[-  c  =  dollars  per  bushel. 

.-.  bx=.  number  bushels  exported. 
/.  a  —  b  x  =  number  bush,  sold  at  home. 
...  (a  —  5  x)  (x  +  c)  =:  number  dollars  home  sale. 
or  (a  —  c  b)  x  —  b  x^  -f-  o,  c  =  number  dollars  home  sale. 

When  this  last  sum  is  greatest  if  x  be  increased,  and  after 
wards  diminished  by  /i,  we  have, 

(a  —  c  b)  h  <  2  b  x  h  +  h  2, 
(a  —  cb)  /»>  2  bxh  —  h*, 

a  —  cb         1501  —  3  X  79 

and  .-.  x  =          -  =  -  —  =  8. 

1b  2  x  79 

Amount  of  sale  $9559. 

It  will  be  perceived  that  the  quantity  ac,  being  con 
stant,  disappears  from  the  reasoning.  In  the  generalization, 
if  cb  ^>  a,  the  value  of  x  would  be  negative. 

9.  A  boy  was   offered   the  use   of  a  rectangular  play 
ground,  which  he  could  surround  with  his  kite  string,  80 
rods  long,  but  he  must  determine  its   dimensions  such, 
should  he  afterwards  incline  to  increase  or  diminish  the 
width  the  least,  it  would  be  at  the  rate  of  enlarging  or 
diminishing  the  area,   10   square  rods  for  one   rod   that 

3 


26  DIFFERENTIAL   CALCULUS. 

should  be  added  to  the  width  of  the  playground,  or  taken 
from  it. 

Let  x  =  width. 

/.  40  —  x  =  length. 
/.  40  ce —  «2  =  area. 

Now,  if  x  become  x  -f-  A,  40  h  —  2  x  h  —  h  2  is  the  area's 
increase ;  but  it  is  conditioned  to  be  equal  to  the  area  of  h 
by  10  rods,  i.  e.,  when  h  •=.  0,  but  when  h  ^>  0  we  must  have, 

10  >  40  —  2  x  —  h. 

Likewise,  if  x  become  x  —  A,  the  area's  diminution  in 
amount  when  positively  expressed,  will  be  40  A — 2ajA-|-/i?, 
which  also  ^>  10  h  by  the  same  condition ;  hence, 

x  >  15  —  $  A, 
and  oj<15  +  jA; 

which  conditions  make  an  equation,  when  h  begins  to  be  an 
amount,  and  while  it  is  practically  0 ;  .:x  =  15.  In  some 
sense  h  may  be  considered  a  mere  suggestion  of  quantity. 

10.  How  great  could  the  playground  ever  be  made  ? 
How  small  ?     Does  it  change  faster  when  near  a  square,  or 
wThen  most  at  variance  with  a  square  ? 

11.  The   number   of  pounds   of  a  certain   commodity 
added  to  its  number  of  dollars  value  per  pound  is  576 ; 
what  is  its  price,  when   the   suggestion   that   the   price 
begins  to  vary  the  least  (in  our  mental  adjustments  of  it), 
requires  the  inference  that  the  value  of  the  whole  will  be 
gin  to  vary  19  times  as  fast? 

12.  A  very  cautious  man  has  been  offered  the  opportu 
nity  of  laying  out  for  himself  a  rectangular  piece  of  ground, 
which  shall  contain  6  acres;  two  contiguous  sides  of  which 
must  agree  with,  and  be  a  part  of  an  established  north  and 
south,  and  an  east  and  west  line,  meeting  like  two  edges  of 
the  sheet  of  this  page.   The  particular  difficulty  has  been  the 


THE  BINOMIAL   SERIES.  27 

establishing  of  the  unspecified  corner.  He  has  walked 
over  the  ground  line  so  much  in  which  he  might  establish 
said  corner,  that  he*  has  worn  a  path  a  portion  of  the  way. 
He  could  not  walk  over  the  whole  track,  because  it  may 
be  found  to  be  infinite  in  length.  He  finally  determines 
the  boundaries  so  that  the  lot  shall  embrace  a  certain 
spring  of  good  water,  and  drives  down  his  stake  for  the 
indeterminate  corner  at  such  a  point  in  that  path,  that 
the  suggestion  of  varying  it  along  that  path,  the  least 
amount,  requires  the  inference  that  the  length  of  his  lot 
must  increase  3£  times  as  fast  as  the  width  would  di 
minish.  Required  the  length  and  width. 

13.  In  the  equation  xy  —  y  —  a  =  0,  a  being  a  con 
stant  x  and  y  variables  ;  if  x  can  be  22,  what  will  be  y,  and 
what  the  tendency  and  rate  of  y  to  increase  or  decrease, 
on  increase  of  x  (if  it  can  increase)  ? 

Ana.  y  to  decrease  2f  £ f  times  as  fast  as  x  to  increase. 


SECTION    Y. 

THE     BINOMIAL     SERIES.  —  SOLUTIONS     BY     INEQUA 
TIONS. 

00.  THEOREM.  If  the  quantity  in  the  Binomial  Series, 
whose  powers  increase,  in  the  successive  terms,  be  sufficiently 
small,  any  term  of  such  series  will  be  greater  than  the  sum 
of  all  that  follow  it,  or  less. 

If  the  binomial  (x  ±  h)n  be  developed  by  this  series,  n 
being  a  whole  number,  negative  or  fractional,  it  becomes, 

xn  ±  A  xn~l  h  +  B  xn~2  A2  ±  Oxn~3  AS  -f  etc.; 
that  is, 

xn  ±  (Axn~l  ±  Bxn~2h-\-  Cxn-5k*  ±  etc.)  A: 


28  DIFFERENTIAL   CALCULUS. 

where  A,  B,  and  C  are  quantities  which  contain  neither 
x  nor  A,  but  are  coefficients  which  may  be  otherwise  com 
posed  of  n  and  numerals. 

61.  Now,  it  is  evident  that  the  first  term  within  the 
parenthesis  will  not  grow  smaller  by  any  diminution  which 
h  may  undergo,  as  all  of  the  succeeding  terms  do  without 
limit,  and  consequently  their  sum,  and  become  less  than 
any  assignable  quantity,  and,  of  course,  less  than  that  first 
term. 

The  above  is  evidently  true  whether  n  be  negative  or 
fractional,  because  we  have  only  to  regard  #,  or  its  recip 
rocal  -,  with  whatever  index,  as  some  quantity. 

63,  If  any  other  term  in  the  foregoing  series,  than  the 
one  just  used,  be  selected,  dividing  it  and  all  succeeding 
terms  by  h  with  whatever  exponent  h  may  have,  the  truth 
of  the  theorem  becomes  apparent.  If  the  selected  term  be 
negative,  it  is  shown  to  be  less  than'  the  algebraic  sum  of 
all  succeeding  ones,  because  that  sum  approximates  zero 
indefinitely  with  A,  while  the  selected  term  cannot  become 
greater.  However,  in  the  application  of  the  theorem,  we 
may  change  the  signs  of  every  term  of  the  series,  as  we 
may  wish  to  compare  quantities  simply  in  their  amount  of 
difference  from  zero. 

The  principle  of  the  theorem  is  true  with  greater  general 
ity  than  the  simple  form  (x  ±  A)'1,  for  any  function  of  #±  A 
developable  in  a  similar  series,  may  replace  that  of  partic 
ular  powers  of  x  ±  A. 

63.  Although  in  a  later  section  technical  definitions  of 
the  terms  maximum  and  minimum  will  be  given,  a  gen 
eral  idea  of  these  values  of  a  function  has  already  been 
given ;  we  proceed  now  to  determine,  by  the  aid  of  the 
theorem,  some  of  these  values  in  problems,  where  a  devel 
opment  gives  rise  to  protracted  series,  and  by  the  use  of 
inequations. 


SOLUTIONS  BY  INEQUATIONS.  29 

PKOBLEMS. 

(V4.  1.  Let  it  be  required  to  determine  whether  the 
remainder  expressed  thus, 

a  x  —  x  3, 
has  a  maximum  or  greatest  value. 

By  inspection,  the  term  -having  the  highest  power  of  x 
is  observed  to  be  negative,  hence  the  remainder  may  evi 
dently  be  infinite  and  negative  ;  and,  the  terms  being  two, 
a  maximum  is  probable. 

Supposing  x  in  the  expression  or  function  of  x,  to  be 
such  in  value  as  to  render  the  function  of  the  greatest 
value,  and  it  be  suggested  that  x  then  receive  the  addition 
of  a  small  quantity  A,  then  the  amounts  appended  to  the 
respective  terms  are  related  thus  : 


i.  e.,  a  <  3  x*  +  (3  x  +  h)  h. 

Again,  if  h  be  subtracted  from  x  in  each  term,  the  amounts 
subtracted  from  each  term,  treated  here  as  positive  for  a 
mutual  comparison  only,  are  related  thus  : 


i.  e.,  a  >  3  a2  —  (3  x  —  A)  h, 

where  if  h  =  0,        a  =  3  a2, 

and  x  =  -4-  (-  V. 

~  W 

Here  we  find  two  answers,  so  we  may  infer  that  we  ought 
to  have  asked  if  a  x  —  x  3  may  not  also  have  an  infinitely 
great  positive  value,  which  we  should  find  to  be  true  in 
the  negative  values  of  x.  When  the  function  is  adapted 
to  express  its  values  with  x  negative,  it  is 

(-J-  a  X  —  aj)  =  —  (—  x  X  —  x  X  —  »), 

that  is,  —  a  x  -\-  a3,  or  x3  —  a  x. 

3* 


30  DIFFERENTIAL   CALCULUS. 

In  reasoning  upon  this  basis,  our  signs  of  inequality  would 
have  been  the  reverse  of  what  they  have  been,  but  the 
result  the  same. 

2.   Let  the  function  of  x  be  ax3  —  x4. 

On  inspection,  the  function  having  different  powers  and 
the  highest  power  of  x  in  a  negative  term,  it  is  probable 
there  may  be  a  maximum  ;  considering  x  to  be  such  that 
the  function  is  at  a  maximum,  and  using  h  as  before,  we 
have, 


and  on  diminution  of  x  by  the  amount  A, 

3  a  x*  —  (3  ax  —  -  a  h)  h  >  4z3  —  (6  x*h  —  4a  A2  +  A3)  A; 


where  we  have  changed  all  the  signs  for  the  reason  before 
given,  and  restored  those  within  the  parenthesis,  because 
they  are  prefixed  by  minus. 

These  inequations  must  remain  such  after  we  have  dimin 
ished  the  smaller  member  by  dismissing  an  added  term 
(3  a  x  +  «  h)  A,  and  have  increased  a  larger  member  by 
dismissing  a  subtractive  quantity  (3  ax  —  ah)  h.  So  that 
representing  the  quantity  within  the  parenthesis  of  the 
right  hand  by  S  and  $',  we  have 

3  aaja4aj3 


hence  3  axz  =  4  x3  .:  x  =  £  a. 

There  is  no  minimum  value  for  ax*  —  a?4,  because  when 
it  is  regarded  with  respect  to  the  negative  values  of  x  it 
appears  as 


—  x4; 


for  the  function^  as  first  given,  is 


SOLUTIONS   BY  INEQUATIONS.  31 

ax3  —  x4; 
i.e.,  a(+xX+xX+x)  —  (+a  X  +*  X  +z  X  +<e). 

Now  the  sign  of  x  changed  gives  us 

—  ax3  —  £c4, 
or,      a(—  xX—  xX—  a?)  —  (—  #X  —  a  X  —  •  #X—  «> 

3.  To  determine  if  the  function  #  aj  3  -j-  a;  4  has  a  minimum 
or  least  value,  at  negative  values  of  x. 

4.  To  determine  if  the  function  3  x3  -|-  7  x  —  5cc2  has  a 
maximum  or  minimum  value,  or  both. 

Supposing  x  to  take  an  increment  A,  and,  whether  we 
regard  the  maximum  or  minimum,  we  find,  after  coupling 
terms  of  like  signs  as  distinct  members  of  an  inequation, 
and  eliminating  as  before, 


which  gives  two  values  of  cc,  one  adapted  to  a  maximum, 
the  other  to  a  minimum. 

5.  To  determine  the  maximum  or  minimum  value  of  the 

function  : 

6  +  z 

--  8  x. 

9  +  z 

Here  in  positive  values  of  x  the  minuend  cannot  be  great 
er  than  1,  nor  less  than  f.  ;  but  the  subtrahend  may  be 
infinitely  negative.  So,  regarding  the  remainder  the 
greatest  possible,  we  must  have,  if  x  be  increased  by  A,  and 
decreased  by  h  : 

5  +  *  +  h        5  +  x 

--  <C    o  fll 


S  +  s  — *        5  +  x-h 

9+*-A~^n^ 8A; 


reducing  to  a  common  denominator,  adding  terms -and 
dividing  by  h,  we  have 


32  DIFFERENTIAL   CALCULUS. 

9  —  5 
(9  +  x  +  A)  (9  +  x)  < 

and 


whence,  when  h  —  0,  we  have 


(9  +  *)« 

which  determines  £,  one  of  the  two  values  of  which  are 
for  a  minimum. 

6.  Has  —  x2  a  maximum  ? 

This  is  equivalent  to  0  —  x2  ;  then  as  before, 

0       2zA       A2 


—  A2, 
.*.  a?  =  0  at  the  maximum. 

7.  Required  the  value  of  #,  which  renders  x4  -\-  ax  a 
maximum  or  minimum. 

It  may,  by  inspection,  be  inferred  to  be  infinitely  great, 
at  one  value  of  aj,  therefore  it  is  probable  there  is  only  a 
minimum  ;  using  zero  to  represent  a  subtrahend,  we  have 
from  , 

x4  +  ax  —  0, 

4  x*  +  h  #-{-«>  0, 
and  4x3  —  h  #'-fa<0; 

...  4  X3  _    a  _  o 


* 

and  the  function  is  at  a  minimum. 


=-© 


SOLUTIONS   BY  INEQUATIONS.  33 

8.  Has  y  a  minimum  in 

y  =  x3  -\-  ax? 

9.  To  determine  whether  y,  the  following  function  of  a-, 
has  a  maximum,  and  at  what  value  of  x : 


i.e,   -H- +  ~ 

'      l+X*       '       1+X2 

Here  we  have  no  visible  expression  of  a  negative  term, 
but  yet,  when  x  is  at  some  greatness,  or  greater  than  1, 
the  first  of  the  resolved  terms  on  growth  of  x  tends  to 
increase,  and  the  second  always  to  decrease  while  x  ^>  0 ; 
but  when  x  in  value  lies  between  zero  and  «J  3  the  study 
of  it  will  show  that  the  reverse  of  the  above  is  true  of 
the  first  term,  therefore  there  is  a  probability  of  a  mini 
mum.  Hence,  when  the  function  is  at  a  minimum,  the 
term  about  to  increase  must  do  that  faster  than  the 
other  diminishes,  so  that 


^   I      **     ^       _£_   I 


—  2xh  +  A2)  (l-f- 


Now  disregarding  denominators,  because  they  are  com- 
mon,  dividing  by  A,  and  using  8  and  8'  as  before,  we 
have, 


34  DIFFERENTIAL   CALCULUS. 

3  a2  _^_  X4  _|_  sfr  >  6  a  +  3  A, 
3  ^2  _|_  3.4  __  #/  h  <  6  a;  —  8  h, 
.-.  3  a;9  +  a4  =  6  JB,  .-.  x  >  1  and  x  <  2. 
When  A  =  0,  the  denominators  become  (1  -|-  x  2)2. 

65.  These  illustrations  have  been  given  in  full,  to  show 
what  course  might   be  adopted  for  the  solution  of  many 
problems.     The  following  sections  will  show  how  the  cre 
ation  of  the  terms  unused,  may  be  dispensed  with,  by  the 
use  of  differentiation. 

Another  advantage  will  be  found  to  be  the  needlessness 
of  any  presumption  in  advance  about  maxima  or  mini 
ma  as  was  apparently  necessary  above.  Further,  the 
successive  maxima  and  minima  of  the  same  function  will 
be  beautifully  deduced  by  the  principles  of  differentia 
tion. 

These  illustrations  have  also  been  extended  to  this  de 
gree,  because  the  deductions  are  purely  algebraic,  point 
ing  out  forcibly  the  necessity  of  differentiation. 

66.  Thus  far  we  have  struggled  for  want  of  the  lan 
guage  of  expression,  which  the  principles  of  differentia 
tion  are  about  to  supply  us,  and  the  mode  adopted,  of 
solution  by  inequations,  will  be  left  as  of  no  further  use, 
and  as  likely  to  be  encumbered  with  insuperable  difficul 
ties. 

Such  are  the  dilemmas  into  which  those  who  slight  the 
calculus  must  find  themselves  involved,  who  would  nev 
ertheless  pursue  its  subjects  of  investigation,  without  its 
technical  methods  and  language  of  expression. 


DIFFERENTIAL  OF   A    VARIABLE.  35 


SECTION  VI. 

DIFFERENTIAL   OF    A    VARIABLE.  —  DIFFERENTIAL   OF 
A  FUNCTION. 

67.  It  may  have  been  seen  in  the  solution  of  some  prob 
lems  that  have  been  presented  that  a  certain  use  was  made, 
of  the  first  one  of  those  appended  terms  of  increase  or 
decrease,  or  rather  in  some  cases,  first  set  of  terms  includ 
ing  in  one  all  those  sub-terms  which  together  are  a  factor 
to  A,  in  composing  one  term  of  the  function  expanded  to 
express  its  new  consecutive  value  —  that  first  term  ap 
pended  to  the  function  in  its  primitive  state. 

Thus  if  the  function  of  x  were 


the  set  of  sub-terms  that  compose  one  as  recognized  in  the 
language  of  the  Binomial  Theorem,  is 


and  the  one  term  is 


the  use  referred  to,  consisted  in  equating  such  term  with  0. 
This  made  an  hypothesis  for  an  inferred  value  of  SB,  and  an 
inferred  characteristic  for  the  function,  while  h  became 
eliminated  when  made  equal  to  0. 

68.  Such  an  expression  is  the  differential  of  the  func 
tion^  while  A,  when  put  at  its  limit,  zero,  in  value,  and  called 
•dxj  is  the  differential  of  the  variable,  where  d  is  a  symbol, 
not  a  quantity,  and  never  has  an  isolated  position. 

The  differential  of  the  function  would  then  need  to  be 
symbolized  as  d  (Fx)^  or  if  y  were  the  function's  repre 
sentative  in  amount,  we  may  put  dy  for  d  (Fx).  The 
expression  Fx  means  a  function  of  x>  hence  F  is  not 
separable  as  a  factor. 


36  DIFFERENTIAL   CALCULUS. 

(iO.  The  expression  dx  has  one  advantage  of  zero  or  0, 
that  it  designates  a  relation  to  x,  in  contradistinction  from 
another  variable  z,  or  w,  which  may  be  associated  in  the 
same  expression  ;  and  from  d  y. 

70.  The  act  of  taking  the  differential  of  a  function  is 
called  Differentiation. 

71.  When  the  function  we  may  wish  to  differentiate,  is 
of  one  term,  not  fractional,  and  has  a  power  of  which  the 
variable  is  the  root,  and  the  index  is  a  whole  positive  num 
ber,  the  Binomial  Theorem  teaches  at  once  that : 

72.  We  should  multiply  together  the  index  of  the  power, 
the  power  having  its  index  diminished  by  one,  the  constant 
factor  if  there  be  such,  and  the  differential  of  the  variable. 

The  product  is  the  differential  required. 

PROBLEMS. 

• 

73.  1.  What  is  the  differential  of  the  function  <e2  of  the 
variable  x,  the  function  being  equated  with  y. 

Ans.  dy  =  %xdx. 

2.  Required  the  differential  of  bx*-=y. 

Ans.  dy  =  %  bx^dx. 

3.  Required  the  differential  of  7  c  x  4  —  y. 

Ans.  d  y  =  28  c  x  3  d  x. 

4.  Required  the  differential  of  150  xn  =  y. 

Ans.  dy  =  150  n  x  n~1  dx. 

5.  Required  the  differential  of  100  x  —  y. 

Ans.  dy=  100  dx. 

It  may  be  readily  inferred  that  if  the  function  has  a  term 
or  terms,  in  which  the  variable  does  not  enter,  such  terms 
are  constant,  and  contribute  nothing  to  the  differential. 
Such  term,  however,  affects  the  function  by  supplying  a 
basis  of  value  common  to  every  value  of  it. 


DIFFERENTIAL   OF   A    FUNCTION.  37 

6.  Required  the  differential  of  10010  x*  -f  5  =  y. 

Ans.  d  y  =  20020  xdx. 

7.  Required  the  differential  of  a  —  5  x  2  =  y. 

Ans.  dy=.  —  10  x  d  x. 

74.  If  we  attend  to  the  origin  of  the  numerical  coeffi 
cients  of  the  second  term  of  the  Binomial  series  (the 
units  of  which  are  a  transfer  of  those  of  the  index),  as  well 
as  to  the  origin  of  such  index,  we  perceive  that 


x'2  =  x.x'  and  d  (#2)  =  2xdx  =  xdx'  -\-x'  dx, 

x3  =  x.x'.x"  and  d  (x3)  =3x*dx  = 

x'  .  x"  dx  +  x  .  x"  dx'  +  x  .  x'  dx", 

where  we  have  placed  accents,  to  denote  the  source  of  the 
different  elements,  and  the  mode  by  which  they  would  con 
tribute  to  the  result,  if  the  quantities  x,  x1,  and  x"  were  not 
alike.  The  condensation  takes  place  because  they  are  pre 
sumed  to  be  alike. 

8.  Let  it  be  required  to  express  in  detail  as  above,  the 
differential  of  x4  or  x  .  x'.  x".  x"'. 

9;  How  then  must  the  differential  of  such  a  function  of 
x  as  y  =;  x  (x  —  a)  be  taken  if  we  do  not  choose  to  per 
form  the  multiplication  ? 

Ans.  dy  =  (x  —  d)  dx  -f-  x  d  (x  —  a). 

10.  Discover  the  identity  of  this  answer  with  the  differ 
ential  of  a;2  —  ax,  which  is  the  above  function  after  multi 
plication  has  been  performed. 

75.  It  is  then  sufficiently  evident  that  to  differentiate  a 
Fx,  which  is  a  product  of  several  functions  of  the  same 
variable  : 

We  should  multiply  the  differential  of  each  factor  by  the 
product  of  the  other  factors  and  add  the  obtained  products 
—  remembering  the  algebraic  meaning  of  the  word  add. 
4, 


38  DIFFERENTIAL   CXLCULtTS. 

1.  Required  dy  from  (a  —  cc2)  (x  —  1)  rrr  y. 

Ans.  dy  = 

2.  Required  d  y  from  (#3  +  a)  (3  a2  +  b)  ==  y. 

Ans.  dy  =  (15  a4  -f  3  5  a2  -)-  6  a  x)  dx. 

3.  Required  d  y  from  £C2  (x  —  a)6  =  y. 

Ans.  dy  =  Zx  (x  —  a)6  dx  -)-  6  a?2  (#  —  a)5  c?x. 

In  differentiating  the  function  5  xt  and  finding  it  to  be 
5  dx^  we  may  observe  that  5x  =  x-^-x-{-x-\-x-\-x,  and 
5dx  =  dx  -\-dx-\-dx-\-  dx-\-dx. 

It  is  almost  self-evident  that  if  we  have  such  a  function 
of  x  as 

ax12  -\-x  —  x^=.y^ 

and  x  take  an  increment  dx,  y  is  affected  by  the  several 
differentials  of  the  terms  which  compose  y.     Hence  : 

70.  To  differentiate  the  one  function  of  a  variable,  which 
is  a  sum  or  difference  of  certain  other  functions  of  it;  we 
must  take  the  differentials  of  the  component  functions  and 
connect  them  by  the  signs  by  which  they  affect  the  one 
function. 

4.  Hence  the  dy  of  ax2  -{-x  —  x*  =  yis 


77.   To  differentiate  a  function  of  a  variable  when  it  is  of 
a  fractional  form. 

5.    Let  _£!_=:  y. 

(*-a)3 

For   convenience  replace   the  numerator  by  N"  and  the 
denominator  by  D,  then 

N 


where  D  y  is  a  product  of  two  quantities, 


DIFFERENTIAL   OF   A    FRACTION.  39 


Le, 

dN—ydD 


D 


replacing  the  value  of  y 

DdN—NdD 


dy  = 


78.  Wherefore,  the  rule  for  differentiating  a  fraction  is 
found  to  be : 

From  the  Denominator  multiplied  by  the  differential  of 
the  Numerator,  subtract  the  Numerator  multiplied  by  the 
differential  of  the  Denominator,  and  divide  the  remainder 
by  the  second  power  of  the  Denominator. 

Or,  by  a  discovery  of  an  artificial  device  for  aiding  the 
memory,  and  simulating  the  form : 

From  denom1  by  differ'-of  numer'  — 

subtra'tf  numeral  by  differen^V  of  denomwa', 

divide  the  remainder  by  (denominator)  2. 

6.  Required  the  differential  of  y  =  — — —  • 

acxdx  —  2  (ax  —  b)cdx 

Ans.  ay  = — . 

C2X3 

79,  If  the  variable  does  not  appear  in  the  denominator 
of  the  function  given,  it  is  evident  that  such  denominator, 
by  taking  unity  as  its  own  separable  numerator,  may  be 
separated  from  the  function  as  a  constant  factor.     The  rule 
results  in  this. 

7.  What  is  dy  in  the  case  of  y  =  x~  =  -  X  ^s? 

a          a 

Ans.  d=^  —  xdx. 


40  DIFFERENTIAL  CALCULUS. 

8.  Required  the  d  y  from  y  =  x~s;   i.  e.,  -j. 

d  (-}=-  —  =-Zx-*dx, 

\X*J  X* 

which  is  in  accordance  with  the  rule  for  positive  indexes. 

80.  Let  it  be  required  to  differentiate  a  function  of  a 
variable,  which  is  at  once  a  power  with  some  root  of  that 
power  expressed  : 

9.  Let  y  =  x$, 
then                                       2/3  —  3.2^ 


,  2xdx        2xdx         2 

dy  =  —  —  =  -  =  - 

3*  Zx\  3 


10.  Next  let 


f*-4*   ,  2     _«    _ 

•  = —dx  = x    *  dx. 

x*  3 


11.  Differentiate  x  n  =y,m  and  n  being  any  whole  num 
bers,  and  the  fraction  — »  positive  or  negative. 


,  m      ll_i    , 

dy  =  —xn      dx. 

n 

m 

12.   Differentiate  x~~»~  =  y. 

7  vn,      _^_j,    7 

dy  = x     n      dx. 

n 

81.   Hence,  generally  to   differentiate  a  function  of  a 
variable,  expressed  by  the  variable  at  a  power  denoted  by 


DIFFERENTIAL   OF   A    FUNCTION.  41 

an  index,  positive  or  negative,  whole   or  fractional,  the 
power  having,  or  not  having,  a  constant  as  factor : 

.Bring  down  that  index,  with  its  sign,  to  be  a  coefficient 
(or  a  part  of  it)  ;  annex  any  constant  which  was  an  ori 
ginal  factor ;  then  the  variable,  having  now  its  index  dimin 
ished  by  unity  ;  then  the  differential  of  the  variable  / — their 
continued  product  is  the  differential  required. 

82.  But  if  the  function  given,  and  having  such  exponent, 
is  not  the  variable,  but  some  function  of  it,  the  above  rule 
still  holds  ;  using,  instead  of  the  word  variable,  that  func 
tion  of  it  which  is  the  root,  for : 

13.   What  isdi/'m  (a  —  x^)^  =  y? 

Using  u  for  a  —  ic2,  du  =  —  Zxdx, 


dy  ==  —  4  x  (a  —  »2)  dx. 
By  the  above  rule  the  differential  of  any  root  is  obtained. 

14.  Required  dy  from  y  •=.  «jx-=x*. 

d* 

83.  From  the  foregoing  we  infer,  that  if  roots  of  powers, 
or  powers  of  roots,  are  expressed  by  fractional  or  other  in 
dexes,  their  differentiation  is  made  plain. 

84.  But  a  root  affected  with  such  index  may  be  some 
function  of  the  variable. 

15.  Required  the  differential  of  y  =  (a  —  XQ)?. 

Putting  u  =  a  —  #2, 
.-.  du  =  —  Qxdx, 

7  7/*\  ^u  xd  x 

4* 


42  DIFFERENTIAL   CALCULUS. 

16.   Required  the  differential  of  the  function  of  &>  viz. 


Ans.  dv  = 


2 

17.   Required  the  differential  of 


(a  — 


SECTION  YII. 
FIRST  DIFFERENTIAL  COEFFICIENT. 

85,  Whenever  a  differential  of  a  function  is  obtained,  it 
will  have  been  seen  that  the  differential  of  the  variable 
(d  £c),  is  always  found  to  be  a  factor  of  it.  If  now  we  divide 
by  this  dx,  its  natural  algebraic  position  is  as  a  denomi 
nator  to  dy  ;  thus,  if  we  have 


d  x 

and  —  will  be  an  expression  in  the  general  form  for  the 

'/  X 

differential  coefficient.  The  actual  quantity  for  the  differ 
ential  coefficient,  in  respect  to  a  particular  function,  will 
be  that  with  which  --  is  equated. 

d  x 

86,    First  Differential  Coefficient  is  the  nomenclature  of 
Leibnitz.     First  Derivative  or  First  Derived  Function  is 


VALUE   OF   FIRST   DIP.    COEF.  43 

by  that  of  Lagrange.  If  the  word  first  is  omitted,  and 
second  or  third,,  etc.,  Is  not  expressed,  the  first  is  under 
stood.  We  use  the  nomenclature  of  Leibnitz,  but  men 
tion  that  of  Lagrange  as  in  use. 

87,  Since  the  value  of  a  ratio  is  properly  expressed  as  a 
quotient  and  by  a  fraction,  —    (meaning   always  by   this 

some  special  quantity  derived  from  an  actual  function)  is 
the  value  of  the  ratio,  between  the  amount  of  the  change 
of  the  function  and  the  corresponding  change  of  the  varia 
ble  ;  more  strictly  it  is  the  limit  of  that  ratio,  as  it  becomes 
when  the  increment  of  the  variable  h  is  made  zero,  when 
we  call  it  d  x. 

The  dif.  coef.  of  a  x3  =  y,  is  3  a  x^  =  d— . 

d  x 

.:dy:  dx  ::  3  a  x*  :  I ; 
that  is,  0  :  0  : :  0  X  3  a  x2  :  0  X  1. 

88.  The  expression  for  a  first  differential  coefficient,  in 

d  y 

its  general  form,  is  —  .  In  a  particular  function  of  one  va 
riable,  the  y  in  dy  is  that  particular  function,  and  the  x  in  dx 
is  the  particular  variable.  The  expresssion  —  is,  in  respect 
to  its'  signification  for  a  value,  the  same  as  -  (and  this 
we  have  already  found  in  algebra  may  have  any  value  what 
soever),  with  this  advantage  over  - ,  that  it  preserves  a  refer 
ence  to  its  origin,  as  a  means  of  determining  its  value. 

8$.  The  value  of  a  fraction  of  which  both  the  numerator 
and  denominator  is  0,  is  determined  by  the  disposition  of 
those  quantities  each  of  which  may  have  actually  become 
0,  to  emerge  from  that  state  ;  then  they  come  into  being 


44  DIFFERENTIAL   CALCULUS. 

with  a  ratio,  the  value  of  which  is  determined  by  their  at 
tendant  factors : 

OXa          0          a 
thus,  — —  =  —  =  — ; 

(c  —  x)  a          a 

or  rather,  -  =  — , 

(c  —  x)  b          b 

which  we  deduce  when  x  has  any  value,  inclusive  of  x  =.  c. 

90.  Let  it  be  required  to  find  the  first  differential  coef 
ficient  of  the  following  functions  of  a  single  variable  x, 
each  standing  equated  with  y  as  the  same  value  with  each 
function ;  but  severally  or  disconnectedly,  the  several 
functions  being  absolutely  independent  of  each  other. 

1.  Given  y  =  152783  x. 

.-.  d-^  =  152783. 

dx 

2.  Given  y  = x  3. 

150000 


d  x         50000 

3.  Given  y=.(bx-\-a)  x5. 

.-.  dJt  =  (6  b  x  -4-  5  a)  x4. 

d  x 

4.  Given  y  =  (17  a  b  —  1)  x  +  34  a  x*. 

.'.  d-?  =  17  a  b  —  1  +  68  a  x. 

dx 

5.  Given  y  =  a  b  x5  —  (a  x  -\-  b  cc2)  5. 


/.       r=  5  a  b  x4  —  5  (a  x  +  b  z2)4  (a  +  2  b  x). 

d  x 


EXAMPLES  OF  FIRST  DIF.  COEFS.  45 

6.   Given  y  =  (a  x  -\-  b  x*  —  c)  2. 

.-.  dJL  =  2  (a  x  +  b  x*  —  c)  (a  +  2  b  x). 


7.   Given  y  =  ^/a  x  -f-  c  a;3. 

sJ»,    _  («  +  3ca;2)c7a;    dy  _        a  +  3  c  z2 


8.   Given  y  =  (a  V  »  +  a2)2  +  a?5  _  (c  ^n  __  j^s. 


9.   Given  y  =  (a  -f-  5  a;)  a?3. 


10.  Given     =—  —  . 


rfy 

da: 
-« i      /T  a  a;2 

11.  Given  v= . 


b  +  x 

dy        (b -\- x)2ax—ax* 


12.  Given  v  =  — — . 


13.  Given  y  = 


,  ~       — 

<^x  2«— 2a 


46  DIFFERENTIAL   CALCULUS. 

14.   What  is  the  first  differential  coefficient  respectively 
of  the  following  (functions  of  x)  =  y  ? 

=y. 


Ans.   -    = 


dx  2 

A.  A/~ 

15.  a  - 


A  dV 

Ans.        = 
«f* 

16. 


x  +  VI 


dy 
Ans.  —  = 


VI— #2(1 +2z  VI— 


17.  (aj  -J~  «)  V  ^  ft2  —  (#  —  «)  2  =  y. 

c?  V 

Ans.  —  = 
18. 


Ans.     ?!=_= 
rfz       2V* 


19. 


Ans.       = 

d  x 


91.  The  reciprocal  of  a  differential  coefficient  of  a  Fx  = 
y  being  evidently  —  in  general  expression,  it  may  be 
found  with  great  facility  in  a  particular  case,  —  it  being 


USE   OP  FIRST   DIF.   COEFS.  47 

necessary,   simply,  to   make   the   quantity  equated   with 
—  a  denominator  of  unity  ;  or,  if  the  dif.  coef.  is  already 

d  X 

in  the  form  of  a  fraction,  to  exchange  numerator  for  de- 

dy 

nominator.  Hence,  whatever  purpose  —  may  serve  in  ref 
erence  to  F  x  =  y,  where  x  is  the  independent  variable, 
and  the  value  of  the  function,  i.  e.,  y  is  the  dependent 
variable,  is  subserved  by  —  in  the  case  in  which  y  is  to  be 
regarded  as  the  independent  variable  of  the  function  whose 
value  is  x,  the  dependent  variable. 


20.  Required  —  in 
dy 


#  V  x  —  — 


-T\.I1S«  - 


SECTION    VIII. 
USE    OF    FIRST    DIFFERENTIAL    COEFFICIENTS. 

92,  We  have  shown  how  a  first  differential  coefficient 
of  any  explicit  function  of  a  single  variable  may  be  ob 
tained,  and  have  deduced  a  general  expression  for  it,  that 

dy 

is,  — ,  which  is  consistent  with  y  assumed  as  the  amount 
d  x 

of  the  value  of  the  function,  with  x  as  the  variable,  and 
with  each  of  their  differentials,  when  made  =  0. 

The  particular  differential  coefficient  is  that  quantity 

with  which  —  is  found  to  be  equated. 

d  x 


48  DIFFERENTIAL   CALCULUS. 

When  a  first  differential  coefficient  contains  the  variable 
ic,  it  has  a  value,  when  a  value  is  assumed  for  x.  Or,  a 
value  may  be  assumed  for  the  dif.  coef.,  within  the  limits 
of  possible  values,  which  may  be  equated  with  the  actual 
particular  dif.  coef.  and  the  corresponding  value  of  x 
deduced. 

This  same  value  of  a;  referred  back  to  the  function,  and 
x  replaced  by  it,  gives  us  also  the  corresponding  value  of 
the  function,  with  which  such  value  of  the  dif.  coef. 
agrees. 

Wherever  we  mention  the  singular  meaning  of  the 
woft'd  value,  of  a  variable  or  a  function,  let  the  plural,  or 
values,  be  understood  in  the  same  connection. 

93.  When  a  first   dif.  coef.  consists  of  several  terms 
(which  we  have  called  sub-terms  in  their  relation  to  the 
binomial  theorem),  connected  by  the  diverse  signs  -f-  and 
— ,  it  must  be  evident  that  the  resultant  sign  for  the  dif. 
coef.  depends  on  the  value  of  that  variable  ;  which,  indeed, 
is  true  in  determining  the  value  of  a  function. 

94.  As  we  have  seen,  a  first  dif.  coef.  may  have  only 
one  possible  value,  i.  e.,  when  it  does  not  contain  the  vari 
able.     It  may  also  have  every  conceivable  value  from  —  co 
to  -|-  co. 

95.  We  shall  use,  as  will  be  seen,  the  differentiation  of 
a  dif.  coef.  for  its  service,  in  investigating  a  dif.  coef.  as  a 
derived  function. 

90.  Sometimes,  in  mathematical  investigations,  we  first 
arrive  at  a  quantity,  which  it  is  necessary  to  regard  as  a 
differential,  or  a  dif.  coef.,  in  a  case  when  the  future  object 
of  search  is  for  its  function  which  we  have  not  possessed ; 
this  search,  when  generalized  for  every  possible  case,  may 
be  very  elaborate,  and  is  denominated  the  Integral  Cal 
culus. 


FOUR  CONDITIONS  OP  VALUE.  49 

97,  We  ought,  in  a  previous  section,  to  have  observed, 
in  the  study  of  functions  in  connection  with  their  vari 
ables,  that  we  have  four  conditions  to  regard  with  respect 
to  positive  and  negative  values. 

1.  The  function  may  be  positive  in  value,  while  the 
variable  is  positive,  thus :  -[-^X  -\-  x  —  ax. 

2.  The  function  may  be  positive  in  value  while  the  va 
riable  is  negative,  thus :  ( —  #)2  =  a?2  or  —  a  X  —  x  =  ax. 

3.  The  function  may  be  negative  in  value  while  the  va 
riable  is  positive,  thus :  ( —  a  X  +  x)  =  —  ax,  or  —  (-}-  x) 

=  —  x. 

4.  The  function  may  be  negative  in  value  while  the  va- 
riable  is  negative,  thus :  («  X  —  #)  =  —  &  x. 

If,  under  the  3d  head,  we  should  have  given  as  an  illus 
tration  —  x3  =  —  (-)-  se  .  -[-  35 .  -f-  ce),  we  remark  that  the 
sign  —  is  the  sign  that  indicates  how  the  amount  a?3  is  to 
be  applied  as  a  value  for  the  function ;  it  is  not  the  sign 
of  the  individual  x. 

Since  x  alone  is  by  no  definition  excluded  from  being  a 
function  of  x,  it  seems  singular  that  —  x  as  a  function  con 
tains  -f-  x  nevertheless  as  a  variable.  But  in  such  use  the 
negative  sign  is  applied  to  x  only  for  its  value  as  a  func 
tion.  Thus,  if  —  x  be  the  function  which  is  also  y,  we 
have, 

y  =  —x, 

that  is,  —  y  =  -\-  x ; 

so  that  the  indication  of  —  x  as  a  function  is,  that  the 
function  has  a  negative  value  to  the  extent  of  -\-x. 

It  is  then  superfluous  to  remark,  that  it  becomes  unne 
cessary  to  use  such  language  as,  a  function  of  —  x,  or 
F  ( —  x),  but  rather,  a  function  of  -f-  x,  F  (-|-  x)  ;  i.  e.,  Fx 
5 


50  DIFFERENTIAL  CALCULUS. 

with  reference  sometimes,  as  it  may  be,  to  negative  values 
of  x  in  it. 

PROBLEMS. 

1.  Required  the  value  of  x  when  it  is  increasing  ¥^  as 
fast  as  27  x  -f  3  x*.  Ans.  3. 

2.  Required  the  value  of  x  when  it  is  increasing  1000000 

2999999 

times  as  fast  as  27  x  -f-  3  x2.  Ans.  —  4  6QOOOOQ- 

A  quantity  is  considered  to  be  increasing   when  it  is 
becoming  a  smaller  negative  one. 

3.  To  determine  the  value  of  x  when  it  is  one  and  the 
same  variable  in  x^   and  a?3,  at  the  values  of  these  func 
tions  when  they  are  increasing  with  equal  rapidity.     Call 
ing  a?2  =  y  and  x3  —  y\  we  have, 

!»  =  *'=  2*  =  8x», 

dx        dx 


4.  Suppose  the  Boston  and  Maine  Railroad  running 
north  from  Boston,  and  the  Old  Colony  and  Fall  River 
Railroad  running  south  from  Boston,  to  be  one  continuous 
north  and  south  railroad  passing  through  Boston,  and 
Worcester  to  be  40  miles  west  of  Boston.  The  cars  on 
one  of  these  roads,  being  30  miles  from  Boston,  are  running 
south  at  23  miles  an  hour  ;  how  fast  are  they  affecting  their 
distance  from  Worcester  ? 

Let        x  =  the  distance  of  the  cars  from  Boston  in  miles. 
2  is  the  distance  from  Worcester,  which  call  y. 


dx  "   (402+2:2)*  " 

x  being  =  30. 


ILLUSTRATIONS   BY   PROBLEMS.  51 

Now,  if  x  vary  23  times  the  natural  x  as  related  to  y,  y 
will  vary  23  X  ±  |  =  ±  13f . 

Ans.  dr  13J  miles.     The  positive  answer  being  adapted 

to  increase  of  distance  from  W.,  the  cars  being  south 

of  B. 

The  ambiguous  answer  is  adapted  to  each  of  the  rail 
roads,  the  direction  of  the  motion  being  the  same. 

5.  At  the  point  where  the  cars,  running  north  32  miles 
per  hour,  are  affecting  their  distance  from  Worcester  18 
miles  per  hour,  how  far  are  they  from  Boston  ? 

d  y  32  a; 

Here,  —  —  -      1  =  18, 

dx  (402  -f  £2)  * 

and  x  —  22.7  miles. 

6.  A  farmer  is  raising  1000  swine,  which,  on  their  attain 
ing  a  certain  weight,  estimated  here  by  their  pork,  he  in 
tends  to  slaughter  and  pack  in  casks,  in  connection  with 
2000  pounds  of  other  pork,  which  is  already  stored  in 
waiting ;    the   cooper,  without   much   consideration,  had 
contracted  to  make  the  casks,  each  so  as  to  contain  one 
animal  and  60  pounds  more ;  but  the  farmer  has  not  de 
cided  at  what  weight  of  growth  to  slaughter  them.    How 
heavy  is  one  when,  allowing  it  afterwards  to  grow,  the 
growth  will  be  at  the  rate  of  one  pound,  while  the  number 
of  casks  required  changes  one  cask  ?  and  will  it  be  increase 
or  decrease  of  the  number  of  casks  ? 

Ans.  Between  180  and  181  pounds;  casks  to  increase. 

7.  Examine  y,  ?/',  and  y"  in  these  three  functions  of  #, 
and  determine  which  never  increases  and  which  never  de 
creases  at  positive  values  of  x ;  and  vice  versa  with  negative 
values  of  x ;  and  which  never  changes  at  either  positive  or 
negative  values  of  x,  and  simplify  the  form  of  one  of  them, 
and  possibly  eliminate  x. 

33  x  +  2000 

o.  ii    = . 

9  a; +  60 


52  DIFFERENTIAL  CALCULUS. 

33^+2000 


9. 
10. 


z  +  60 

34  x  +  2000 

a; +  60 


11.  Does  one  of  the  above  expressions,  and  which,  fail 
to  be  a  function  of  ar,  in  such  *ense  that  the  change  of  x 
produces  any  effect  on  the  value  of  the  expression,  which 
is  to  fail  entirely  ? 

12.  A  man  having  4320  dollars,  purchased  a  horse,  and 
with  the  remainder  of  his  money  purchased  sheep,  at  such 
rate,  that  the  money  expended  for  a  horse  would  buy  as 
many  sheep  as  they  were  worth  dollars  apiece.     How, 
with  the  preservation  of  this  relation,  would  the  number 
of  sheep  purchased,  tend  to  vary,  compared  with  the  num 
ber  of  dollars  paid  for  the  horse,  if  he  pays  for  him  36  dol 
lars,  while  in  the  contemplation  of  paying  a  greater  sum 
than  that  ? 

Ans.  The  number  of  sheep  will  diminish  lOyg^  times 
as  much  of  a  sheep  as  of  a  dollar  more  that  should 
be  paid  for  a  horse,  or  at  the  rate  of  lOyf^  sheep  for 
one  dollar  additional  paid  for  a  horse. 
When   a   coefficient  has   the    ambiguous   sign,  we  are 
often  able,  as  here,  to  prefer  one  to  the  other  from  positive 
knowledge.     The  cost  of  a  sheep  is  not  to  be  supposed  a 
negative  sum  of  dollars. 

13.  A  boy  was  holding  by  a  cord  in  his  hand  a  horse, 
the  cord  attached  at  the  horse's  mouth  and  held  upon  the 
level  ground,  thus  permitting  the  horse  to  eat  the  grass 
upon  a  circle  of  ground;  but  having  given  the  animal  90 
feet,  he  commences  drawing  the  cord  in  at  the  rate  of  3 
feet  per  second.   How  many  square  feet  per  second  was  the 
circular  plot  of  grass  diminishing?  that  plot  which  the  con 
ditions  allow  the  animal,  if  he  were  sufficiently  active  to 
avail  himself  of  it. 


ILLUSTRATIONS   BY   PROBLEMS.  53 

14.  The  boy  begins  to  climb  a  tree  5  inches  per  second, 
and  to  let  out  cord  4  inches  per  second.    How  fast  per 
second  is  the  horse's  circle  changing  in  square  feet,  when 
the  boy  gets  10  fedt  high  and  has  let  out  70  feet  of  cord? 

15.  While  climbing  as  above,  and  when  10  feet  high, 
how  fast  should  he  pay  out  cord  that  the  circle  may  re 
main  stationary  ? 

16.  A  ship  is  sailing  north-west  at  15  miles  an  hour;  at 
what  rate  is  she  gaining  in  north  latitude  per  hour  ? 

Ans.  10.601  miles. 

17.  On  the  discharging  of  coal  from  a  vessel,  it  is  raised 
high  in  the  air  and  thrown  down  into  heaps,  which  may  be 
cones.      On  one  occasion,  the  height  of  the   heap   was 
always  f  of  the  diameter  of  the  base ;  when  its  height  was 
15  feet,  and  the  solid  contents  were  increasing  at  the  rate 
of  160  cubic  feet  per  hour,  how  fast  per  hour  was  the 
height  increasing  ? 

18.  How  fast  per  hour  was  the  height  increasing  when 
it  had  become  32  feet  high,  the  conic  heap  gaining  160 
cubic  feet  per  hour,  and  being  always  a  similar  heap  ? 

19.  A  boy,  amusing  himself  by  throwing  stones  into  a 
pond  of  still  water,  to  see  the  circle  of  waves  expand,  per 
ceived  that  the  diameter  of  one  circle  increased  3  feet  per 
second  when  it  had  become  24  feet  in  diameter;  how  fast 
was  the  area  increasing  in  square  feet  at  that  instant  ? 

20.  It  is  required  to  divide  70  into  two  such  parts,  that 
the  suggestion  of  increasing  one  part  afterwards  at  the 
expense  of  the  other,  would  implicate  an  increase  of  their 
product  40  times  as  fast  as  that  one  part  should  increase. 

Ans.  15  and  55 ;  the  15  to  increase. 

21.  At  a  pin  factory,  a  certain  number  of  pins  are  stuck 
in  a  row  in  the  papers ;  3  more  than  that  number  of  rows 
are  put  in  a  paper,  and  one  less  than  that  same  number  of 
papers  are  put  in  a  package.     It  being  suggested  to  di 
minish  the  number  of  pins  in  a  row,  how  does  that  qualify 


54  DIFFERENTIAL  CALCULUS. 

the  change  of  the  number  in  a  package,  while  there  hap 
pens  to  be  16  in  a  row? 

There  is  this  disability  in  this  problem :  the  idea  of  frac 
tional  pin  cannot  be  entertained,  and  the  change  of  an 
entire  pin  cuts  off  the  application  of  the  principle  of 
the  initial  ratio.  Those  quantities,  a  part  'of  which  can 
be  practically  considered  as  of  the  same  nature  as  the 
whole,  are  best  adapted  to  investigation  and  study  by  the 
calculus. 

The  disability  is  not  mathematical,  but  concerns  that 
practical  economy  by  which  events  and  things  are  sub 
mitted  to  calculation. 

22.  Some   boys   rolling  a  spherical   ball   of  snow,  ob 
served  that  when  it  was  28  inches  in  diameter,  it  was 
increasing  in   diameter  5  inches  per  minute ;  how  many 
cubic  inches  per  minute  were  its  solid  contents  increas 
ing? 

23.  A    certain    cook   adopted   200    handfuls    and    300 
ounces  of  flour,  to  be  made  into  cakes,  of  £  of  such  hand 
ful  each;  if  this  would  make  1840  cakes,  what  disposition 
had  the  number  of  cakes  to  vary,  if  the  number  of  ounces 
in  a  handful  were  diminished  the  least  amount. 

Ans.  Cakes  increase  in  number  at  the  rate  of  24  to  the 
first  ounce  that  should  be  diminished  from  the  hand 
ful,  the  weight  of  the  cake  of  course  diminishing. 

24.  A  glazier  prepared  a  quantity  of  putty  sufficient  to 
set  100  panes  of  glass,  with  an  excess  of  180  ounces;  but 
at  this  stage,  receiving  orders  to  set  glass  of  such  size  of 
pane  as  to  require  per  pane  more  putty  than  his  first  esti 
mate,  it  is  required  to  determine  how  the  number  of 
panes,  which  he  may  be  able  to  set  with  the  whole  of  the 
putty,  will  vary  with  the  increase  of  the  number  of  ounces 
requisite  for  one  pane  :  1st,  to  determine  this  by  a  general 
expression ;  and  2d,  what  the  amount  of  the  ratio  is,  in  the 


ILLUSTRATIONS   BY   PROBLEMS.  55 

particular  case,  when  he  was  to  use  3  ounces  in  setting 
his  first  panes. 

2d  Ans.  Panes  diminish  20  times  as  fast  in  number  as  the 
putty  per  pane  increases  per  ounce,  i.  e.,  while  3  oz. 
suffice  per  pane  in  the  first  estimate. 
25.  Will  y,  or  the  sum  of  the  following  quotients,  in 
crease  or  decrease,  when  7  is  replaced  by  a  numerical 
quantity  just  larger,  y  being  a  function  of  7  ;  7  temporarily 
supplying  the  place  of  x  ? 

25     .    7 


26.  A  person,  thinking  to  propound  a  puzzle,  said  he 
was  in  the  habit  of  purchasing  the  article  A,  at  the  rate 
of  one  dollar  per  ounce,  and  on  each  such  occasion,  of  pur 
chasing  also  the   article  B,  paying  the  same  amount  of 
money  for  the  whole  of  the  article  B  as  for  the  whole  of 
A.     Now,  on  every  occasion  of  these  two  associated  pur 
chases,  the  weight  of  the  amount  purchased  of  A,  added 
to  that  of  .2?,  made  20  ounces.     But  it  is  not  intended  that 
on  all  occasions  the  weights  of  A  were  the  same,  but  that 
they  varied  indefinitely.     On  the  occasion  when  he  may 
have  purchased  3£  ounces  of  A,  if  we  proceed  to  consider 
the  occasion  when  he  may  have  purchased  any  the  least 
more  of  A,  it  is  required  to  determine  how  we  are  to  find 
the  price  per  ounce  of  B  to  change  between  the  corre 
sponding  occasions. 

Ans.  Number  of  ounces  of  A  to  increase  Tf  ¥  as  fast  as 
the  price  in  dollars  of  B  per  ounce. 

27.  It  is  required  to  determine  the  price  per  ounce  of 
the  article  B,   on   an  occasion  when,  on   comparing  the 
price  with  that  of  the  same  article  on  another  occasion,  in 
passing  to  which  there  may  have  been  an  increase,  the 
least  possible,  in  price,  the  corresponding  increase  in  the 
amount  of  A  purchased  must  have  been  equal  to  it  ;  the 


56  DIFFERENTIAL   CALCULUS. 

ounces  of  the  A  and  the  dollars  per  ounce  of  13  being  com 
pared  numerically  in  units.  Ans.  3  ry?y  dolls. 

28.  There  are  two  numerical  quantities,  x  and  y,  and 
they  are  such  that  5000  times  x  plus  3  times  y,  are  always 
equal  to  200  times  x  times  y.     It  is  required  to  determine 
if  x  can  have  the  value  1500,  in  which  case  it  is  required 
to  determine  the  corresponding  value  of  y. 

Ans.  2/  =  25^f^T. 

29.  It  is  required  to  determine  how  x  and  y,  at  the  above 
values,  are  disposed  to  change  their  values. 

Ans.  y  will  be  disposed  to  diminish  ?ouu^E8i7TyF  as 
as  x  to  increase. 


SECTION  IX. 

SUCCESSIVE  DIFFERENTIATION.  —  SECOND,  THIRD,  ETC., 
DIFFERENTIALS. 

98,  When  a  first  differential  coefficient  contains  the 
variable,  it  is  evidently  a  function  of  it,  called  first  derived 
function.  When  the  variable  x  of  the  primitive  function 
varies,  the  differential  coefficient  will  itself  vary.  Hence 
those  suppositions  in  some  problems,  where  a  differential 
coefficient  contains  cc,  and  is  supposed  or  inferred  to  have  a 
value,  give  correct  results  only  for  an  exactly  specified,  or 
exactly  inferable,  value  of  a?,  and  of  the  function. 

Where  a  certain  rate  has  been  inferred  for  the  cars  to 
affect  their  distance  from  Worcester,  as  at  so  many  miles 
per  hour,  the  execution  of  it  could  not  take  place  during 
a  minute  or  an  entire  rod.  The  supposition  and  the  infer 
ence  are  good  for  only  an  instant  of  time,  and  at  a  point 
only  in  place  ;  at  the  succeeding  instant  of  place  and  time, 


SUCCESSIVE   DIFFERENTIATION.  57 

the  differential  coefficient  has  changed,  the  distance  from 
Worcester  has  changed,  and  the  rate  of  the  changing  of 
that  distance  has  changed.  (Page  50,  Prob.  4.) 

What  is  more  natural,  then,  than  to  employ  the  same 
means  in  determining  the  general  character  of  the  derived 
function,  that  we  already  partially  have,  in  regard  to  the 
primitive — differentiate  it?  and  why  not,  perhaps,  continue 
to  differentiate  the  second  derived  function,  or  second  dif 
ferential  coefficient  ? 

Let  it  be  required  to  differentiate  successively  y  =  15  a?3, 
and  while  doing  it,  to  evolve  the  notation  for  these  acts ; 
and  for  the  results 

we  have  y  =  15  #3, 

,.!>:  =  45  »«, 

dx 

i.  e.,  —  Xdy  =  tt>x^ 

dx 

where  —  is  a  factor  in  the  notation  — ,  and  should  be  sup- 

d  x  dx 

posed  constant,  not  only  because  we  are  able,  in  case  of  a 
fraction  which  is  to  change  its  value,  to  throw  that  change 
entirely  into  the  numerator  or  denominator  at  will ;  but 
because  we  set  out  in  this  case  with  a  function  y,  which 
would  change  if  x  did,  and  we  suppose  x  to  change  by  an 
amount  which  we  can  exactly  define,  h  or  d  #,  with  the  ex 
pectation  of  throwing  all  quantity  that  must  have  any  other 
nature  than  an  exact  and  certain  one,  which  may  be  arbi 
trarily  made  uniform,  upon  the  change  of  y.  Hence,  dif 
ferentiating  45  a;2,  and  expressing  that  of  its  equal  in  the 
notation,  we  have 

—  X 

d  x 

We  evidently  can,  and  ought  to,  assume  the  independent 
variable  to  vary  uniformly,  thus  allowing  the  results  to  be 


58  DIFFERENTIAL  CALCULUS. 

all  made  manifest  in  the   value   of  the   function   which 
depends  upon  it. 

If  now  we  agree,  as  is  the  custom,  to  represent  d  (d  y) 
the  differential  of  the  first  differential  of  y  by  d*  y,  where 
the  2  does  not  signify  a  power,  but  the  second  act  of  dif 
ferentiation,  we  shall  have 

-1  X  d*  y  —  90  x  dx  and  ^  =  90  as, 

dx  dx2 

where  d  cc2  is  by  proper  algebraic  act  the  second  power  of 
dx,  and  is  not  the  same  as  d  («2). 

In  a  similar  manner  may  third,  fourth,  and  more  differ 
entiations  be  performed  on  a  function  that  admits  of  them, 

and  the  notation  —  -  and  —  -   etc.,  be  derived. 

dx3  dx^ 

In  the  last  function,  viz.,  15  x3, 

£1»  =90,^  =  0,  etc. 

dx3  dx* 

Hence,  differentiations  will  terminate  of  functions  in 
which  the  variable  appears  at  a  power  denoted  by  a  whole 
and  positive  index,  and  when  the  function  is  not  of  a 
fractional  form,  the  variable  in  a  denominator.  In  such 
cases  differentiation  may  never  terminate. 

1.   Differentiate,  successively,  6  #3  —  5  a;2  -[-60  x  =  y. 

d-?-  —  18  a;2  —  10  x  +  60,  ^  =  36  x  —  10  ; 
dx  dx* 

i.  e.,  -L  X  d*  y  =  36  x  —  10, 


=  36,^  = 

dx*  dx* 


SUCCESSIVE  DIP.    COEFS.  59 

2.  Differentiate  (a  —  x)  *J~x  +  V17  —  0,  a  .F  (a,  y)  =  0  ; 
i.  e.,  a  function  of  x  and  y  =  0. 


3.  Given  4  w2  +  w  =  y,  in  which  u  is  this  Fx,  viz.,  w  = 
3  a2,  to  find  —. 

d  x2 
d2  y 

4.  Required  ~  in  6  z3  —  5  a;2  +  60  x  +  10  ==  y. 

Ans.  36  x  —  10. 

5.  When  the  sub-terms  are  reduced  to  a  resultant  term, 
what  is  the   sign   of  36  x  —  10   when  x  >  3  ?     When 
*<i? 

6.  Given  28  aj»  —  y2  =  0,  to  find  Fx  =  y,d—  and  —  . 

dz  dz2 

Ans.      =  x     ~28    -  =     ~28 


a2  d2  v 

7.  Given  a;2  +  —  (2  b  x  —  a2)  =  y,  to  find  -~, 

Ans.  2(l— f,  I 
*  \ 

8.  Given  24  a;2  —  y3  +  10  x  =  0,  to  find  —. 

9.  Given  a;6  —  2  a;3  y  +  y2  —  a;2  =r=  0,  to  find  d*y 


Ans.  6  x. 
10.   Given  4 £c2-rccr=rs,  and  3  22-|~2s  =  y,  to  find — -. 

d  x2 

The  following  analogy  is  worthy  of  note :  If  we  take  the 
third  powers  of  the  natural  numbers  1,  2,  3,  etc.,  and  then 


60  DIFFERENTIAL  CALCULUS. 

their  differences,  and  then  the  differences  of  these  differ 
ences,  and  so  on,  we  have 

0         1         8        27        64        125         216        343,    etc., 
1        7        19       37        61          91         127,    etc., 
6       12       18        24         30          36,    etc., 
6666  6,    etc., 

0000,    etc. 
If  we  take  y  =  x3  and  differentiate,  we  have 


dx* 


dx' 


dx* 

If  a  first  dif.  coef.  for  any  determining  reason  has  the 
value  0,  we  may  determine  what  the  function  is  about  to 
do,  in  regard  to  increase  or  decrease,  by  the  sign  of  the 
second  dif.  coef.,  which  is  an  important  principle.  If  the 
second  dif.  coef.  =  0,  we  determine  it  by  the  sign  of  the 
third,  and  so  on. 

The  reason  why  a  dif.  coef.  may  have  the  value  0,  is  ow 
ing  to  an  aggregate  of  sub-terms  with  different  signs,  which 
may  compose  it,  and  a  particular  value  of  the  variable  also 
supposed.  Or  if  there  be  but  one  term,  and  it  contain  the 
variable  as  a  factor,  it  =  0  when  x  =  0. 


TAYLOR'S  THEOREM.  61 

The  process  of  successive  differentiations  may  terminate, 
or  never ;  some  fractional  forms  are  of  the  latter  character, 
as  well  as  those  with  negative  and  fractional  indexes. 

Differential  coefficients  may  change  their  signs  as  the 
variable  is  traced  through  successive  values.  Of  course 
the  general  form  of  notation  cannot  show  this.  Particular 
values  of  the  variable  determine  this  change  of  sign. 

When  a  dif.  coef.  does  not  contain  the  variable,  it  may 
not  have  the  value  zero. 

To  determine  whether  a  dif.  coef.  can  have  the  value 
zero,  we  equate  it  with  zero  and  reduce  the  equation. 


SECTION  X. 
TAYLOR'S  THEOREM. 

99,  It  is  the  purpose  of  Taylor's  Theorem  to  lay  down, 
in  a  general  expression,  in  the  form  of  a  series,  when  this 
is  possible,  the  different  orders  of  differential  coefficients, 
with  their  signs  and  necessary  factors  attached,  according 
to  which  (primarily)  a  function  of  a  single  variable  is  de 
veloped,  with  an  increment  or  decrement  to  the  variable, 
and  in  the  order  of  the  increase  of  the  integral  powers  of 
the  increment  or  decrement,  by  the  natural  series  of 
numbers. 

Let  y  =  F  x  and  Y=  F  (x  -\-  h\  and  assume 
Y=  y  +  Ah  +  Bh*  +  Ch*  +,  etc., 


where  A,  .Z?,  (7,  etc.,  are  "quantities  which  x  or  constants 
can  express,  and  which  are  now  wanted  to  replace  A,  £, 
(7,  etc.,  with. 

6 


62  DIFFERENTIAL   CALCULUS. 

Remembering  that  x  -|-  h  is  the  new  variable,  and  that 
we  can  ascribe  a  new  variation  of  x  -(-  h  to  either  x  or  h 
at  pleasure,  while  the  other  of  the  two  will  be  constant,  we 
have,  by  differentiating,  first,  with  respect  to  A, 

—  =  -4  +  2.Z?fc  +  3  <7A2+,  etc.; 

'/  h 

next,  with  respect  to  x, 

d  T        dy    .    d  A  ,     .     d  B  d  C  , 

—  =  —  -f-  —  ^  +  —  A2  +  —  A3  +,  etc. 

rfa;          d  x          d  x  d  x  d  x 

But  the  differential  coefficients  of  like  powers  of  h  are 
identical,  because  the  same  thing  has  been  differentiated  in 
each  case. 

A         dy      T>         dA       n         dB 
Hence,  A  =  -^-,  J3=  —  ,   C7=  --  ; 

dx  Idx  3dx 


thatis,    A  =  -9  B=-.-,  C=-.  -JL,eto. 

dx  dx2     2'  dx3      2.3 

Hence  we  have  Taylor's  Theorem, 


. 

rf*      1  rfa:2      1.2         rfa;3      1.2.3 

If  A  be  negative  in  the  original  F  (x  -\-  A),  the  signs  of 
the  terms  having  the  odd  powers  of  h  will  be  negative,  in 
general  expression,  but  such  terms  may  have  &  particular 
positive  value. 

1.  Place  F  x  =  y,  namely,  15  x  -\-  x*2  —  y,  into  Taylor's 
Theorem,  or  as  it  will  be  when  x  becomes  x  -j-  h. 

Ans. 

-+(2)— 

2.  Place  this  F  x  =  y  in  Taylor's  Theorem,  viz.  : 


TAYLOR'S  THEOREM   APPLIED.  63 

5«  +  19  a;2  —  3  x*  =  y. 


(38  — 18  «)  —  +  (— 18)     h* 

1.2  1.2.3 

where,  on  account  of  the  factors  A,  etc.,  no  term  but  the 
first  has  actual  greatness  when  h  =  0,  but  an  initial  or 
relative  greatness.  Now,  since  multiplying  a  function  or  a 
coefficient  by  a  constant,  does  not  aifect  the  greatness  of 
the  variable  in  its  relation  to  any  laws  or  principles  of 
change  in  the  function  or  coefficient,  or  to  its  sign  in  the 
resultant  of  a  term,  we  are  able,  for  most  practical  purposes, 

to  dispense  with  the  factors  — ,  — ,  etc.,  in  the  use  of  this 

theorem.  And  it  is  in  this  sense  that  we  may  have  made 
an  implied  use  of  the  theorem  without  a  formal  demonstra 
tion.  The  coefficients  as  factors  of  each  term  have,  how 
ever,  in  themselves  real  amounts. 

In  noticing  the  example  above,  we  might  hesitate  about 
the  significance  of  -f-  ( —  18),  but  it  is  of  course  =  —  18  ; 
hence  an  important  notice  that  the  plus  of  a  general  nota 
tion  may  be  reversed  by  special  considerations.  In  the 

d  i/           d  ^  v 
_  and above,  i.  e.,  in  their  equivalent  in  the  particular 

dx  dx2 

quantities,  the  actual  sign  of  the  resultant  of  a  term  cannot 
be  known  without  hypothesis  for  the  value  of  x9  although 
the  notation  gives  -)-  before  the  general  term. 

3.  Place  (a  —  x)  3  =  y,  in  Taylor's  Theorem,  in  its  state 
which  has  just  been  passed  by  in  consequence  of  the  last 
growth  of  x.  Making  h  the  decrement : 

F(x-h)  = 

'  "   I  "\"T     <.a         ''i.j          '1.2.3' 


64  DIFFERENTIAL   CALCULUS. 

4.  Do  the  same  with  x 5  =  y. 

5.  Do  the  same  with  —  x  3  =  y. 

6.  Present,  in  Taylor's  Theorem,  the  variable  taking  an 
increment,  some  terms  of  the  function  of  ic,  viz. : 


7.  Present,  in  Taylor's  Theorem,  a  few  terms  of  — ^  =  y. 

8.  The  same  required  of  -  —  =  y. 
Taylor's  Theorem  has  failing  cases. 

100.    Says  Professor  Playfair: 

"A  single  analytical  formula  in  his  (Brook  Taylor's) 
Method  of  Increments  has  conferred  a  celebrity  on  its 
author  which  the  most  voluminous  works  have  not  often 
been  able  to  bestow.  It  is  known  by  the  name  of  Taylor's 
Theorem,  and  expresses  the  value  of  a  variable  quantity  in 
terms  of  the  successive  orders  of  increments. 
If  any  one  proposition  can  be  said  to  comprehend  in  it  a 
whole  science  it  is  this,  for  from  it  almost  every  truth  and 
every  method  of  the  new  analysis  may  be  deduced.  It  is 
difficult  to  say  whether  Taylor's  Theorem  does  the  most 
credit  to  the  genius  of  its  author  or  the  power  of  the  lan 
guage  which  is  capable  of  concentrating  such  a  vast  body 
of  knowledge  in  a  single  expression.  This  Theorem  was 
first  published  in  1715." 


THEORY   OP  MAXIMA   AND   MINIMA.  65 

SECTION  XI. 
THEORY   OF  MAXIMA  AND   MINIMA. 

101,  The   most  marked   characteristic   of   a   function, 
whether  of  one  or  more  variables,  is  a  maximum  or  minimum 
value  of  it  ;  it  is  so  marked  as  at  once  to  arrest  earnest 
attention  ;  but  yet,  it  is  only  an  incidental  or  particular 
condition  among  its  general  values,  or  in  its  general  na 
ture. 

102,  A  function  is  at  a  maximum  value,  or,  in  abridged 
language,  is  at  a  maximum,  when  it  has  such  value  as  will 
be  diminished  in  its  nearest  and  earliest  change,  if  the 
variable,  having  the  value  answering  to  that  condition,  be 
either  increased  or  diminished  the  least  amount  ;  so  that 
we  must  have 

Fx>  F  (x  +  h), 

and  Fx^>F(x  —  h\ 

103,  A  function  is  at  a  minimum  when  it  has  such  value 
as  will  be  increased  in  its  earliest  and  nearest  change,  if 
the  variable  be  either  increased  or  diminished  the  least 
amount  ;  so  that  we  have 


and 

Here  we  may  observe  that  we  have  in 
F(x-  h\ 
Fx, 

and  F  (x  +  h), 

6* 


66  DIFFERENTIAL   CALCULUS. 

three  values  of  a  function  indicated  in  an  indeterminately 
close  rank  of  succession,  and  that 

x  —  A, 


and  x  -)-  A, 

are  the  same  and  corresponding  values  of  the  variable. 

We  are  referring  to  but  one  function,  in  the  use  of  the 
expressions  F  (x  —  A),  F  x,  and  F  (x  -f-  A),  and  since  we 
speak  of  F  x  as  at  a  fixed  value,  we  should  rather  say 
F  (a  —  A),  F  a,  F  (a  -|-  A),  meaning  thereby  three  values 
of  a  function  of  a  single  variable,  such  as  belongs  to  it  for 
x  =  a  —  A,  then  x  =  a,  then  x  =  a  -f-  A  ;  but  in  this  we 
are  using  a  varying  x.  Now  F  (x  —  A),  F  #,  F  (x  -\-  A), 
are  abbreviated  modes  of  asserting  the  same  thing,  with  a 
now  fixed  or  unvarying  x. 

With  the  above  explanation,  we  use  se,  etc.,  instead  of  a, 
etc.,  for  the  better  preservation  of  associations  connected 
with  Taylor's  Theorem.  By  this  theorem  we  have  the 
general  developments  or  expansions  for  F  (x  —  A)  and 
f  (x  -j-  A)  ;  between  which,  however,  we  will  arrange  F  x 
as  an  intermediate  condition,  with  the  value  y  ;  they  are, 


,  eta. 


dx     1         dz2     1.2        dx*      1.2.3 

2.  Fx  =  y. 

3.  F  (x  +  A)  == 

„  +  **.*+*'_*.  J^  +  *!*._^-+,etc. 

1     d*      1      'dx2      1.2    'da:3      1.2.3 

Now  since,  as  in  the  case  of  the  Binomial  Theorem  (60), 


THEORY  OP  MAXIMA   AND  MINIMA.  67 

to  which  the  above  are  similar  in  respect  to  being  series 
with  the  powers  of  A  ascending,  h  may  be  taken  so  small 
that  the  first  or  any  term  having  h  as  factor,  must  be 
greater  than  the  sum  of  all  succeeding  terms,  such  term 
may  be  reasoned  upon  as  in  itself  determining  whether  y 
is  indicated  as  increased  or  as  diminished  by  that  same  any 
term,  and  all  its  sequel,  without  our  paying  any  regard  to 
terms  succeeding  such  one. 

Concerning  the   factors   of  the  several  successive  dif. 

hz          h3 

coefs.,  viz.:  A,  —  ,  -  ,  etc.,  we  may  remark,  that  since 

h  itself  is  supposed  to  be  indeterminately  small,  we  need 
not,  in  reference  to  the  purpose  answered  by  A,  A2,  etc.,  be 
solicitous  about  any  differences  between  them,  nor  whether 
either  have  as  a  divisor  1,  or  2,  or  6,  or  24,  etc. 

If,  then,  Fx  >  F  (x  —  h)  and  Fx  >  F  (a  +  rt),  as  in 


the    case  of  F  x  =  max.,   then   neither  --  .  -    nor 

d         7  d  x      \ 

-|  --  -  -  can  indicate  increase,  which  they  would  not  do 

d  x      1 

if  these  dif.  coefs.  were  =  0,  for  then  their  signs  become 
of  no  force. 

Now  we  may  be  able  to  find  the  value  a?,  if  it  has  one, 
which  verifies  the  following  equation 


dx 

Our  general  formulas,  then,  when  —  =  0,  after  eliminat 
ed  X 

ing  this  dif.  coef.,  become, 


dx*      1.2         dx*      1.2.3 
5.   Fx  =  y. 


6. 

dx*      1.2         dx*      1.2.3 


68  DIFFERENTIAL   CALCULUS. 

dz  y 
Now  it  is  evident  that  if  F  x  =  a  max.,  —   must  be 

d  x2 

negative,  because  there  ought  not  to  be  indicated  or  real 
ized  any  increase  of  y,  in  equations  4  and  6,  F  x  or  y  being 
by  supposition  the  greater.  If  we  test  this  second  dif. 
coef.  by  applying  to  it  the  value  or  values  of  x  already 
found,  and  find  that  it  is  negative,  we  shall  already  have 
verified  a  maximum  for  F  x. 

This  second  dif.  coef.,  or  any  other,  may  have  a  particular 
value,  the  reverse  of  its  sign  by  general  notation,  for  its 
value  must  depend  on  that  of  #,  and  consequently  the  lia 
bility  must  be  incurred  of  its  value  passing  through  zero, 
the  sub-terms  of  such  dif.  coef.  having  of  themselves  -|-  or 
—  signs. 

d*y 

If,  nevertheless,  we  find  this  -   -   to  be  positive,  as  it 

d  «c2 

should  be  in  case  Fx  is  a  minimum,  when  of  course 
F  x  <  F  (x  —  h)  and  F  x  <  F  (x  —  A),  then  we  have 
verified  a  minimum  for  F  x. 

But  the  case  may  occur  when,  after  verifying  its  value, 
we  must  find 


dx* 

in  such  case  we  find  the  following  to  be  expressive  of  this 
condition  : 

7. 


d3y          A3         .     d*y  h* 


.  . 

y  -L.  _     .   --  L  —     . 

1 


dx3      1.2.3         dx*      1.2.3.4 


THEORY  OF  MAXIMA   AND   MINIMA.  69 

Now  it  is  evident  that  F  x  cannot  be  a  maximum  or 

d*y  d3y 

a  minimum  unless  —  —  and  +  -  -  have  the  same  sign, 
d x3  d x3 

i.  e.,  concur  in  indicating  decrease  of  y  in  case  of  a  maxi 
mum,  or  increase  in*  case  of  a  minimum,  and  they  cannot 
concur  unless  each  =  0,  and  then  no  amount  is  indicated ; 
so  this  third  dif.  coef.  becomes  eliminated,  and  we  proceed  to 

d*  y  d2  y 

-  with  the  same  remark  with  which  we  approached  — 

dx*  dx* 

and  should  approach  all  even  dif.  coefs. 

104.  In  order,  then,  to  determine  the  value  or  values  of  x, 
which  render  any  proposed  function  a  maximum  or  mini 
mum,  we  must  deduce  the  value  or  values  of  x  from  putting 

—  =  0,  and  substitute  such  value  or  values  successively  in 
d  x 

the  succeeding  dif.  coefs.,  until  we  arrive  at  one  which  does 
not  vanish,  i.  e.,  become  zero  /  if  this  one  be  the  %d,  5th, 
1th,  9th,  etc.,  the  value  we  have  found  will  not  render  the 
function  either  a  maximum  or  minimum  /  but  if  the  dif. 
coef.  not  vanishing,  be  the  2d,  ^th,  6th,  etc.,  it  will ;  the 
function  being  at  a  maximum  if  the  dif.  coef.  proves 
negative  ;  at  a  minimum  if  it  proves  positive. 

105.  Since  the  variable  may  be  found  to  have  more  than 

one  value  deduced  from  —  —  0,  the  function  may  have 
d  x 

more  than  one  maximum,  or  more  than  one  minimum,  or 
any  number  of  each  :  in  which  case,  if  the  function  is  con 
tinuous,  they  may  alternate  in  consecutiveness.  In  such 
case  a  maximum  may  be  found  for  one  value  of  x,  and  a 
minimum  for  another  value  of  it. 

106.  It  is  obvious,  on  reflection,  that  when  a  constant  is 
common  to  every  term  of  a  function  as  factor,  such  factor 
may  be  disregarded,  in  determining  the  value  of  tc,  at 
which  a  maximum  or  minimum  occurs. 


70  DIFFERENTIAL   CALCULUS. 

The  following  considerations  are  obvious  and  important 
in  abridging  the  method  of  finding  maxima  and  minima :  — 

107.  Radical  signs   and  indexes,  affecting   collectively 
every  term  of  a  function,  may  be  disregarded. 

108.  There  is  no  maximum  or  minimum  of  the  function 
when  x  is  infinite,  because  it  has  no  succeeding  value,  nor 
when  y  is  infinite,  because  it  cannot  be  diminished  by  a 
finite  quantity. 

In  an  early  section  of  this  treatise,  the  rationale  of  maxima 
and  minima  was  the  most  inductively  demonstrated,  with 
reference  to  selected  functions;  but  there  remained  the 
advantages  of  formal  instead  of  virtual  differentiation,  and 
the  use  of  its  nomenclature,  to  be  pointed  out  in  this  sec 
tion,  if  we  would  aim  at  the  most  explicit  rules. 

109.  A  variable  a*,  on  which  a  function  y  depends,  may 
have  its  own  maxima  and  minima,  which  may  be  found 
without  the  necessity  of  determining  what  F  y  —  a;  by 
deducing  the  value  of  x  from  the  reciprocal  of  the  first  dif. 
coef.  of  the  function  put  =  0,  or  from  the  first  dif.  coef. 
put  =  co. 

110.  When  a  function  is  of  a  fractional  form,  and  has 
its  variable  only  in  its  denominator,  it  is  evident  that  the 
maximum  value  of  the  function  corresponds  with  a  mini 
mum  value  of  the  denominator  considered  as  a  certain 
other  function,  and  vice  versa. 

111.  If  the  function  is  fractional  in  form,  and  the  de 
nominator  constant,  it  is  evident  that  the  reciprocal  of  that 
denominator    is   the    constant  factor   to   the   numerator. 
Hence,  such  a  product  is  greatest  when  its  variable  factor 
is  greatest,  or  the  numerator  alone. 

112.  When  a  dif.  coef.  is  in  the  form  of  a  fraction,  it  is 
evident  that  it  must  have  the  value  zero,  when  its  nurnera- 


THEORY   OF   MAXIMA    AND   MINIMA.  71 

tor  has  that  value,  if  at  the  same  time  its  denominator  does 
not  also  reduce  to  the  value  zero. 

113,  Since  it  is  always  necessary  to  know  the  sign  of  a 
denominator  in  determining  the  value  of  a  fraction,  we  may 
remark  that  when  the  denominator  of  a  dif.  coef.  is  an  un 
disturbed  second  or  greater  even  power,  that  denominator 
is  always  positive. 

PROBLEMS. 

Place,  in  Taylor's  Theorem,  this  function  of  a  single 
variable  tc,  namely,  10  x  —  x'2  =  y,  first  with  an  incre- 

h    h* 

ment,  second  with  a  decrement,  — ,  — ,  etc.,  being  under 
stood  as  successive  factors  of  each  term  after  the  first. 

F  (x  +  h)  =  (10  x  —  x  2)  +  (10  —  2  x)  +  (—  2)  +  0 ; 
F  (x  —  h)  —  (10  x  —  x*)  —  (10  —  2  x)  +  (—  2)  —  0 ; 
also,  F  x  —  (10  x  —  #2)  ±  0. 

Now  if  F  x  is  at  a  maximum  or  minimum,  or  first  say 
if  it  is  at  a  maximum,  then  neither  -\-  (10  —  2  a?)  nor 
—  (10  —  2  x)  can  indicate  increase  of  it ;  which  they 
would  not  do  if  10  =  2  x,  or  what  is  the  general  state- 

dy 

ment  (by  the  notation  of  the  theorem)  if  —  =  0. 

J2  ^  "^ 

Now  the  — -  being  —  2  in  both  cases,  it  intimates  de- 

d  x2 

crease  in  both  cases,  when  its  factor  is  allowed  the  same  or 
as  real  a  suggestion  of  being  something  as  is  suggested  for 
the  presumed  change  of  the  function's  value. 

Examine,  as  above,  x%  —  10  a?  for  indications  whether 
it  has  a  maximum  or  minimum. 

dz  y 

But  —  might  contain  the  variable  and  vanish,  L  e., 
equal  0. 


72  DIFFERENTIAL   CALCULUS. 

rf3  y 

In  such  cases  —  takes  the  place  in  the  continued  reason- 

d  ^  *u 
ing  of  the  first,  and  takes  the  place  of  the  second, 

and  so  on. 

114,  It  is  required  to  determine  at  what  value  or  values 
of  x,  and  of  y,  the  following  (functions  of  x)  r=r  y  have 
maxima  or  minima,  if  they  have  such. 

1.  y  —  25  x  —  x2.  Ans.  y  a  max.  when  x  = 

And  when  y  = 

2.  y  =  a  x  -\-  b  x3.  Ans. 

3.  y  =  XQ  —  x.  Ans.  y  =  min.  when  x  = 

And  when  y  = 

4.  y  =  3  x*-\-t>.  Ans. 

5.  y  =  a?2.  Ans.  cc  =  min.  when  a;  =  0. 

And  when  y  =  0. 

6.  y  ==  3  a3  —  54  a;2  -f  315  x  +  5000. 

Ans.  y  =  max.  when  cc  =  5,  y  being  =  5600. 
±zr  min.  when  cc  =  7,  y  being  =  5588. 

7.  v  =  — ^— .  Ans. 

a;2-^* 

2 

8.  y  r=  jc2  X  (b  —  x).       Ans.  y  =  max.  when  x  =  —  b. 

3 


9.  y=  —  X  2  ^ax  —  x*. 

a  .  3 

Ans.  y  =  max.  when  x  =  —  a. 

4 

And  when  y  = 

10.  y  =  6  as  —  a2.  Ans.  y  =  max.  when  a;  = 

And  when  y  = 

*«        4*3  +  2a  Ans<  y  =  min.  when  x  =  a*. 

5  x2  And  when  y  =  f  X  #*• 

12.  y  =  60  +  x3  —  3  a  x*  -)-  3  a2  aj  —  a3. 

dy  d3  y 

In  this  instance,  if  —  =0,  x= a,  but  when  x==a,      ,=0, 

da;  ax* 


PROBLEMS   IN   MAXIMA   AND   MINIMA.  73 

d3 y  d3 y 

and =  —  6,  and  -I ==  4-  6 ;  so  that,  since  there 

d  x3  d  x'* 

is  not  a  concurrence  of  the  values  of  the  third  dif.  coef.  in 
sign,  and  this  third  dif.  coef.  cannot  =  0  in  accordance 
with  the  rule  which  is  here  sustained,  there  can  be  no 
maximum  nor  minimum. 

13.  y  =  (x  —  a)  4.  Ans. 

14.  y  =  (x  —  a  —  b  —  c)2n,  n  being  a  whole  number. 

Ans. 

15.  y  =  (b  —  a;)  2n  +  *,  n  being  a  whole  number. 

Ans. 

16.  y=(b  —  x)  5.  Ans. 

17.  y  =  b+(x-a)*. 

This  is  a  case  of  an  exception  to  the  rule,  for  all  the  dif.  coefs. 
become  infinite  when  x  =  a,  and  y  is  then  =  b.  There  is 
a  maximum  when  x  =  a,  because  F  x  >  F  (x  —  h)  and 
F  x  ^>  F  (x  +  7t),  which  may  be  verified  by  algebraic 
methods,  because  Taylor's  Theorem  fails.  But  if  the  ex 
ponent  be  greater  than  1,  and  less  than  2,  and  its  denomi 
nator  odd,  —  will  not  be  infinite  in  a  function  of  which  the 

d  x 

root  is  x  —  a,  at  the  value  of  x  =  a,  but  there  will  not  be 
a  maximum  or  minimum  becau 
nary  for  its  negative  sign  ;  as  in 

18.  y  =  6 +(^  —  0)4; 
where  y  has  no  max.  or  min. 

19.  (a.)  How  great  can  y  =  8  x  —  a?2  be? 

Ans. 
(b.)  How  small  can  y'  =  80  +  a2  —  10  x  be ? 

Ans. 
(c.)  What  is  the  value  of  x  if  we  put  y  =  y'? 

Ans.  Imaginary. 
7 


a  maximum  or  minimum  because  ±  —  becomes  imasi- 

dx*  & 


74  DIFFERENTIAL   CALCULUS. 

Can  we  then,  indiscriminately,  by  hypothesis,  make  any 
two  functions  of  the  same  variable  equal  to  each  other ; 
that  is,  their  difference  =  0  ? 

20.  Has  the  variable  x  a  minimum  in  y  =.  (ce2  —  a2)'  ? 

Ans. 

21.  y  =  (ax  —  x*)-1. 

22.  y  =  20  +  (6  —  x)  *. 

23.  How  great  is  x  while  positive,  and  y  is  increasing 
the  fastest  in 

2/:=30a:  +  180  x*  —  20  tc3? 
Make  —  a  maximum,  or         =  0,  and  determine  x. 

24.  y  =  (x  —  a)  5. 

25.  y  =  (x  —  b)s. 

26.  y  =  x'1  —  7  cx*  +  21c*x5  —  35  c3cc4  +  35  c4^3  — 
21  cs  x*  +  7  ce  x  +  175  a  b. 

27.  y  =  (x  -}-  a)6.      Ans.  A  minimum  when  x  =  —  a. 

28.  y^^  +  c)?. 

The  above  functions  are  offered  to  bring  into  use  and  to 
verify  more  of  the  conditions  of  the  demonstration  with 
regard  to  dif.  coefs.  after  the  second,  than  is  commonly  re 
quired.  They  will  show  the  utility  of  Taylor's  Theorem, 
as  the  foundation  of  the  demonstration,  to  be  remarkable. 


PROBLEMS  IN   MAXIMA   AND   MINIMA.  75 


SECTION  XII. 

PROBLEMS  FURNISHING  EXPLICIT  FUNCTIONS  OF  ONE 
VARIABLE  ;  FOR  DETERMINING  THEIR  MAXIMA  AND 
MINIMA. 

115.  This  section  will  present  a  collection  of  problems. 
It  is  not  to  be  expected  that  every  value  of  a  function  or 
variable,  which  may  be  algebraically  determined,  can  have 
a  rendering  or  practical  use  within  the  conditions  of  any 
problem  ;  or  that  the  conditions  and  elements  of  the  prob 
lem  can  be  restated  for  accommodation  of  all  such  alge 
braically  determined  results.  In  articles  6  and  7  we  found 
that  this  is  not  possible  ;  we  shall  have  repeated  occasions 
to  verify  the  same  impossibility. 

Of  the  four  conditions  of  value  for  a  function  in  rela 
tion  to  those  of  the  variable  mentioned  in  article  97,  we 
shall  find  that  generally  not  more  than  one  is  available  for 
any  significance  within  the  conditions  of  the  practical 
economy  of  such  problems ;  but  occasionally  two  are. 


1.  (a.)  The  present  problem  is  a  common  algebraic  one : 
A  and  J5  set  out  from  two  towns,  distant  247  miles  from 
each  other,  and  travelled  the  direct  road  till  they  met.  A 
went  9  miles  a  day,  and  the  number  of  days  at  the  end  of 
which  they  met,  increased  by  the  number  of  miles  B  went 
per  day,  was  31.  Required  the  number  of  miles  B  went  a 
day.  Ans.  23.36  miles,  or  —  1.36  miles. 

Here  the  negative  result  must  be  rejected,  for  although 
it  might  be  executed  as  miles,  it  evidently  could  not  as 
days. 


76  DIFFERENTIAL  CALCULUS. 

(&.)  The  same  problem  as  for  the  calculus :  A  and  J?  set 
out  from  two  towns,  distant  247  miles  from  each  other,  and 
travelled  the  direct  road  till  they  met.  A  went  9  miles  a 
day,  and  the  number  of  days,  at  the  end  of  which  they 
met,  increased  by  the  number  of  miles  1$  went  a  day, 
was  the  least  possible,  on  the  conditions.  Required  the 
number  of  miles  B  went  a  day.  Ans.  6^. 

(c.)  Required  the  number  of  days  at  the  end  of  which 
they  met,  increased  by  the  number  of  miles  £  went  a  day, 
as  by  the  last  condition.  Ans.  15.65. 

2.  A  certain  company  at  a  tavern  had  a  reckoning  of 
143  shillings  to  pay,  but  4  of  the  number  being  so  un 
generous  as  to  slip  away  without  paying,  the  remainder 
settled  the  bill  after  the  landlord  had  thrown  off  10  shil 
lings  from  the  amount,  and  it  W'as  found  that  the  original 
company  was  such,  or  within  a  fraction  of  such,  that  a 
payer's  portion  was  increased  the  greatest  amount  it  could 
be  for  any  number  for  the  original  company,  greater  than 
four.     Required  the  number  as  diminished  by  any  such 
fractional  man. 

3.  (a.)  A  man,  F,  is  one  of  a  crew  of  a  returned  fleet 
of  fishing  vessels,  which  are  the  same  in  number  as  the 
men  of  the  like  crews  of  each  vessel ;  and  he  receives  his 
equal  share  of  960  dollars  of  bounty  money.     His  own 
crew  with  himself  expend  24  dollars  for  dining  together. 
It  is  required  to  adjust  the  number  of  men  to  a  crew,  for 
the  condition  that,  by  this  receipt  and  disbursement,  F  finds 
the  least  money  left  as  his  own. 

Ans.  80  men ;  F  pays  15  cents  more  than  his  share  of 
the  bounty. 

(b.)  It  is  required  to  adjust  the  number  of  men  for  a 
crew  for  the  condition  that  the  bounty  money  should  just 
pay  F  and  his  companions,  if  he  had  any,  for  their  dinner ; 
also  required  the  price  of  the  dinner  to  F  for  this  condi 
tion. 


PROBLEMS   IN   MAXIMA   AND   MINIMA.  77 

(c.)  It  is  required  to  describe  the  fleet  in  number  of  men 
and  vessels,  if  it  may  be  called  a  fleet,  for  the  condition 
that  F,  after  paying  for  the  dinner,  conies  off  with  936 
dollars. 

4.  A  certain  pin  factory  has  on  hand  ready  for  sale,  ex 
cept  in  the  packing,  3,200,000  rows  of  pins  as  they  are 
commonly  stuck  in  papers.     From  these  pins  100  boxes 
are  to  be  made  up,  if  possible,  for  sending  away,  each  box 
containing  packages,  each  package  containing  papers,  and 
each  paper  rows,  each  row  pins,  the  same  number.     Now 
the  number  of  individual  pins  not  included  in  this  lot  was 
the  greatest  possible.     Required  the  entire  number  made  ; 
the  number  to  be  sent  away,  the  number  that  will  remain, 
and  the  number  in  a  row. 

Ans.  In  part,  64,000,000 ;  20  in  a  row. 

5.  Required  the  numerical  quantity  which  exceeds  its 
second  power  the  most.  Ans.  ^. 

6.  Required  the  numerical  quantity  of  which  twice  the 
second  power  exceeds  thrice  its  third  power  the  most. 

Ans.  f. 

7.  Two  vessels,  A  and  B,  were  freighted  each  with  500 
or  a  tons  of  coal.     On  the  passage,  the  vessel  A  having 
sprung  a  leak,  a  certain  number  of  tons  were  transferred 
from  A  to  B ;  both  cargoes  were  sold  for  one  35th  (one 
£>th)  as  many  dollars  per  ton  as  there  were  tons  so  trans 
ferred.     Now  the  mutual  product  of  the  number  of  dollars 
of  the  proceeds   of  the   two   unequal    cargoes  was   the 
greatest  possible.     Required  the  number  of  tons   trans 
ferred.  Ans.  353  -J-  tons. 

The  function  of  the  variable  which  is  put  at  the  maxi 
mum  is 


and  it  will  be  seen  that  b  exercises  no  power  in  determining  x. 


78  DIFFERENTIAL    CALCULUS. 

8.  A  speculator  expended  900  dollars  in  the  purchase  of 
animals  at  an  equal  price  each.  After  including  with  this 
lot  15  other  animals  purchased  at  the  same  cost  each,  at  a 
subsequent  time,  he  sold  all  for  1150  dollars,  at  equal  rates 
each,  when  he  found  that  he  had  gained  the  greatest  possi 
ble  profit  on  each  animal.  Required  how  many  at  first 
were  purchased,  and  the  profit  on  each. 

Let  x  =  number  purchased, 

and  y  =  dollars  profit  on  each. 

1150     900  _ 
"  x  +  15    ~x 
250  x  —  13500 


X*  -f-15  x    250 

Ans.  y  and  y1  are  at  a  maximum  at  the  same  value  of  cc, 
viz.,  x  =  115.4,  the  number  of  animals  required. 

9.  Required  the  area  and  each  side  of  the  greatest  right 
angled  triangle  which  has  the  sum  of  its  hypothenuse  and 
base  18  inches. 

10.  Required  how  a  rectangle  must  be  restricted  when 
it  contains  the  greatest  area  for  a  constant  sum  of  the  four 
sides.  Ans.  It  must  be  a  square. 

11.  Required  the   sides  severally  of  the  largest  right 
angled  triangle  of  which  the  perimeter  is  22  in  any  units 
of  length. 

12.  (a.)  A  certain  reservoir,  containing  an  unspecified 
quantity  of  water,  is  receiving  153  casks  of  water  per  day; 
9|-  times  that  cask  full  is  distributed  per  day,  to  each  of 
as  many  families  as  the  cask  holds  gallons.     This  state  of 
receipt  and  distribution  is  continued  45  times  as  many  days 
as  the  measure   holds  gallons.      How  many  gallons  did 


PROBLEMS   IN   MAXIMA   AND   MINIMA.  79 

the  cask  contain,  in  case  it  was  known  that  the  reservoir 
gained  and  retained  the  most  water  possible. 

Ans.  11 75^  galls. 

(b.)  If  the  cask  is  supposed  to  be  of  some  size,  and  then 
to  grow  larger,  in  the  play  of  many  suppositions,  how  great 
is  it  when,  increasing  at  a  fixed  rate,  it  implicates  the 
greatest  gain  of  the  reservoir's  water.  Ans.  5 f-f  galls. 

(c.)  Of  what  influence  is  the  45  in  problem  (a.)  in  the 
matter  of  influencing  the  above  results  ?  Ans.  None. 

13.  (a.)  A  farmer  has  a  triangular  plain  situated  be 
tween  three  crossing  public  highways,  so  barren  and  pointed 
at  one  angle  that  it  is  not  worth  his  while  to  make  a  fence 
enclosing  all  that  point.  The  sides  are  81,  74,  and  15  rods. 
He  concludes  to  run  a  straight  fence  on  the  side  15  rods 
long,  and  on  portions  of  the  other  sides  nearest  this  short 
one,  and  then  run  a  fence  across  the  lot  parallel  to  the  side 
15  rods  long.  It  is  required  to  determine  the  length  of 
this  cross  fence  and  the  area  of  the  fenced  lot  and  of  the 
unfenced  lot,  if  all  the  fence,  in  number  of  rods  long, 
bears  the  least  possible  ratio  to  the  number  of  square  rods 
fenced. 

(b.)  Releasing  now  the  condition  that  the  straight  cross 
fence  must  be  parallel  to  the  side  15  rods  long ;  required 
its  length  when  the  above  ratio  may  be  still  smaller,  and 
the  smallest  possible  ;  then  how  many  rods  of  the  81  and 
of  the  74  are  respectively  fenced. 

(c.)  Required  to  determine  if  some  portion  of  every 
possible  plane  triangle  may  not  be  cut  off  by  a  straight 
line  and  at  each  of  its  angles,  and  the  law  of  all  such 
lines,  the  perimeters  being  thus  reduced  to  a  minimum 
ratio  in  units  of  any  name  to  the  square  units  of  area,  for 
the  six-sided  figure  thus  produced,  i.  e.,  preserving  some 
portions  of  the  original  sides  in  position. 

Ans.  The  lines  must  all  be  tangent  to  the  greatest  in 
scribed  circle. 


80  DIFFERENTIAL   CALCULUS. 

(d.)  Required  to  determine  the  nature  of  an  original 
triangle  when  the  hexagon  thus  produced  may  be  the 
largest  possible  part  of  it.  Ans.  An  equilateral  triangle. 

14.  (a.)  A  farmer  has  a  lot  of  land  in  the  shape  of  a 
right  angled  plane  triangle,  of  which  the  hypothenuse  is  a, 
the  other  sides  b  and  c.     Required  to  determine  the  length 
and  breadth  of  the  largest  rectangular  lot  which  can  be 
laid  out  in  it. 

(b.)  The  sides  of  a  plane  triangle,  not  necessarily  right 
angled,  are  respectively  given,  it  is  required  to  inscribe 
three  largest  parallelograms,  the  half  of  each  as  determined 
by  a  diagonal,  being  one  identical  portion  of  the  original 
triangle. 

(c.)  And  it  is  required  to  find  the  largest  rectangle  which 
can  be  inscribed  in  the  above  triangle,  and  having  the  equiva 
lent  of  its  own  half  in  common  with  the  halves  of  them. 

15.  Divide  a  into  two  such  parts  that,  one  part  multi 
plied  by  the  second  power  of  the  other,  shall  be  a  minimum. 

16.  If  coin  is  conditioned  to  be  cylindric  in  shape,  which 
it  always  seems  to  be,  how  must  it  be  proportioned  that  the 
greatest  bulk  may  be  united  with  the  least  surface,  and 
consequently  the  liability  to  wear  away  by  friction   the 
least  for  a  cylinder. 

Ans.  Diameter  of  face  and  thickness  equal. 

17.  A  turner,  having  the  trunk  of  a  locust  tree  in  the 
shape  of  the  frustum  of  a  right  cone,  such  that  if  the  por 
tion  having  the  apex  were  restored,  it  would  complete  a 
cone  28  feet  long  in  the  axis  and  16  inches  in  diameter  at 
the  base,  is  told  to  turn  out  the  largest  cylindric  gate-post 
possible,  whatever  the  length  of  the  frustum  might  be. 
Required  to  determine  what  length  of  it  he  would  use,  and 
the  diameter  of  the  post  required. 

18.  (a.)  A  turner  is  given  a  lignum-vita?  ninepin-alley 
spherical  ball  4  inches  in  diameter,  and  told  to  turn  out  the 
largest  possible  right  cone.     Required  its  dimensions ;  and 


PROBLEMS   IN   MAXIMA    AND   MINIMA.  81 

the  weight  of  the  ball  being  uniform,  required  to  deter 
mine  the  weight  of  his  chips  or  waste,  as  compared  with 
that  of  the  cone.  Ans.  Height  of  cone  2f  inches. 

(b.)  It  is  required  to  determine  whether  a  section  of  such 
a  ball,  passing  through  the  axis  of  such  cone,  shows  a  maxi 
mum  triangle  in  a  circle,  —  a  problem  which  may  be  deter 
mined. 

19.  («.)  A  milkman,  M,  delivered  daily  in  New  York 
some  quarts  of  milk  to  each  of  some  customers,  on  each  of 
some  streets,  the  quarts,  customers,  and  streets  being  the 
same  in  number ;  and  in  addition  he  delivered  85  quarts  in 
Brooklyn.     Now  N,  another  milkman,  hearing  of  this,  said 
he  believed  that  himself  furnished  daily  each  of  his  own 
customers  8  more  quarts  than  Hfdid  his,  which  M admitted  , 
that,  although  they  were  all  on  Chatham  Street,  both  agree 
that  N  had  3  times  as  many  customers  as  M  had  on  any 
street;  JV  was,  therefore,  ready  to   wager  that  his  own 
deliveries  of  milk,  in  the  aggregate,  were  the  greatest.     The 
discussion    grew  warm,  and  they  put  the  statements  in 
writing  as  above.     Admitting  the  claims  of  the  latter,  or 
giving  N  every  advantage  in  the  interpretation  of  numbers, 
does  he  win  ?     Giving  M  every  advantage,  how  much  may 
his  daily  aggregate  exceed  JV's  ? 

(b.)  The  next  day  M  acknowledged  that  his  delivery  in 
Brooklyn  amounted  to  only  72  quarts.  How  does  the 
original  bet  stand  on  this  hypothesis  ? 

(c.)  What  is  the  difference  of  the  aggregate  deliveries 
of  the  two  men  (on  the  hypothesis  of  the  85  quarts  deliv 
ered  in  Brooklyn)  when  they  are  the  nearest  alike  ?  and 
who  owns  the  difference  ?  And,  on  this  hypothesis,  how 
many  quarts  do  each  deliver  in  all  ?  and  how  many  quarts 
to  a  customer  ? 

20.  (a.)  A  hound  starts  to  catch  a  hare  which  is  22,000 
feet,  in  their  common  line  of  motion,  in  advance  of  him  ; 
from  which  point  the  hare  makes  off  157  times  as  many 


82  DIFFERENTIAL   CALCULUS. 

leaps  as  there  were  feet  in  each  leap ;  during  the  same 
time  the  hound  makes  after  the  hare  2714  leaps,  each  of 
the  same  length  as  the  hare's.  Now,  such  was  the  length 
of  the  leaps  that,  when  they  were  completed,  the  hound 
was  brought  nearer  to  the  hare  than  had  the  leap  been  of 
any  other  length  whatever ;  what  was  its  length  ? 

Ans.  8 iff  feet. 

(b.)  Could  the  hound  have  caught  the  hare  by  any 
length  of  their  common  leap?  Or  could  the  function  of 
the  leap,  which  expresses  the  final  distance  of  the  animals, 
=  0? 

(c.)  How  far  apart  were  they  on  completing  the  leaps 
$|o  T.  fee£  long  ?  How  far  had  each  run  ? 

(d.)  Had  both  started  when  the  hare  was  2200  feet  in 
advance,  might  there  have  been  two  distinct  lengths  of  the 
leap,  adopting  either  of  which  the  hare  would  have  been 
caught,  and  in  what  two  places  ?  And  what  length  of  leap 
would  have  put  the  hound  most  in  advance  of  the  hare  ? 

(e.)  A  hound  starts  to  catch  a  hare,  which  is  22,000  feet 
in  advance  of  him,  and  the  hound  makes  159  times  as 
many  leaps  as  there  are  feet  in  each ;  while  the  hare,  start 
ing  at  the  same  instant,  makes  2814  leaps,  each  of  the  same 
length  as  the  hound's.  Adjust  the  leap  to  the  greatest 
success  or  gain  of  the  hare.  Distance  apart  then,  and  the 
travel  of  each,  what  ? 

(/.)  Supposing  the  animals  move  uniformly,  and  during 
the  same  time,  according  to  a  rate  that  is  to  produce  a  re 
sult  of  distance  specified  in  problem ;  when  the  hare  has 
leaped  100  rods,  where  is  the  hound  ? 

21.  (a.)  An  incendiary,  escaping  arrest,  travelled  by  rail 
road  cars  3  f  hours  at  a  certain  rate,  when  an  accident 
happening,  tending  to  delay  the  cars  indefinitely,  he  resorts 
to  horses  upon  a  highway  by  the  side  of,  and  continuous 
with,  the  railroad.  He  thus  rides  at  one  third  the  rate  at 
which  he  had  gone  by  cars,  and  as  many  hours  as  leagues 


PROBLEMS  IN   MAXIMA   AND   MINIMA. 

per  hour,  and  successively  on  as  many  horses  as  hours  on 
each,  when  he  judges  it  best  to  conceal  himself.  The  next 
day  an  officer  travels  in  pursuit  from  the  same  point  of 
starting,  and  proceeds  6  times  as  far  as  any  one  horse  had 
carried  the  fugitive,  and  7£  leagues  more,  and  the  pursuit 
ends ;  now,  whether  he  arrives  at  the  locality  of  the  fugi 
tive  or  not,  may  depend  upon  the  rate  at  which  the  latter 
travelled,  say  by  cars,  per  hour.  Required  the  rate  or  rates 
when  the  officer  accomplished  as  much  distance  as  the 
fugitive. 

(&.)  In  case  that  by  any  rate  of  travel  of  the  fugitive  by 
cars,  we  find  the  pursuer  accomplished  as  much  distance  as 
the  fugitive,  it  is  required  to  determine  which,  by  any  pos 
sibility,  may  have  accomplished  the  more  distance,  and  how 
much  more. 

(c.)  Required  the  limits  of  rate  of  travel  of  the  fugitive 
by  cars,  by  which  he  accomplishes  the  more  distance ;  also 
the  limits  by  which  the  pursuer  accomplishes  the  more. 

x3        6x2    .    35  x 
Ans.  At  three  different  values  of  x  in  -  —  - 

27  9  9 

7^  =  y,  is  y=  0  ;  at  x  =  5,  y  is  at  a  max. ;  at  x  = 
7,  y  is  at  a  min. 

22.  The  square  root  of  a  certain  numerical  quantity  is 
taken   from   57,  and  the   remainder  multiplied   by   that 
square  root,  and  the  product  is  a  maximum.     Required  the 
quantity.  Ans.  812£. 

23.  A  and  JB  own  1500  square  rods  of  land,  and  also  5 
equal  square  lots  lying  together,  and  they  propose  to  divide 
all  their  land  between  themselves.     In  consideration  of  the 
quality  of  the  land,  A  agrees  to  accept  and  J2  to  grant  a  lot, 
a  side  of  one  of  the  lots  in  one  dimension  and  100  rods  in 
the  other.     Whence  .7?'s  share  was  found  to  be  the  smallest 
tract  he  could  possibly  have,  for  any  size  of  those  lots  what 
ever.     Required  the  dimensions  of  _Z?'s  share,  and  of  one 
of  the  lots.  Ans.  _Z?'s  share  1000  square  rods. 


84  DIFFERENTIAL   CALCULUS. 

24.  There  arc  two  level  lanes,  which  are  straightly  and 
perpendicularly  walled  at  each  of  their  sides  ;  their  widths 
are  respectively  13  and  17  feet.     They  meet  at  right  an 
gles.     A  straight  pole,  of  which  the  diameter  may  be  called 
nothing,   is  to   be   carried  level  past  this  corner  on    the 
shoulder  of  a  boy.     Required  the  length  of  it  when  it  is 
the  longest  possible.  Ans. 

Here  the  minimum  length  of  pole  is  the  logical  maxi 
mum,  in  the  economy  of  the  problem. 

25.  A  right  cone  is  one  of  which  the  axis  is  perpendicu 
lar  to  the  base.     The  base  of  a  right  cone  is  8  inches,  its 
height  is  14  inches.     Required  the  diameter  of  the  largest 
sphere  which  it  can  enclose. 

26.  (a.)  How  far  apart  must  a  person,  whose  feet  are  b 
or  10  inches  long,  place  the  foremost  end  of  his  feet,  while 
his  heels  are  together,  that  the  area  of  the  base,  on  which 
he  may  be  said  to  stand,  may  be  the  largest,  and,  therefore, 
the  most  secure  as  a  general  support? 

(b.)  But  since  his  heels  cannot  in  strictness-be  placed  on 
one  point,  it  is  required  to  determine  the  above  question, 
with  the  allowance  that  his  heels  may  be  a  or  15  inches 
apart,  but  we  will  now  condition  that  the  feet  be  symmet 
rically  situated,  writh  reference  to  the  line  which  joins  the 
heels. 

The  solution  of  this  problem  will  embrace  the  previous 
one  if  we  put  a  =  0. 

Let  x  —  the  distance  required 


then,  x=  -±V2#2H • 

The  ambiguous  sign  of  the  above  result  indicates  that  the 
principle  involved  does  not  discriminate  that  the  weight 
of  the  body  bears  on  the  heels  more  than  on  the  toes,  and 


PROBLEMS   IN   MAXIMA    AND   MINIMA.  85 

so  tolerates  the  condition  that  the  foremost  ends  of  the 
feet  may  be  nearest  together,  as  well  as  the  heels,  which  is 
true. 

27.  A  company  of  90  men  was  formed,  in  1849,  for  min 
ing  in  California,  on  equal  shares,  but  before  actually  com 
mencing  labors  they  are  induced  to  admit  more  members 
into  the  partnership.  After  working  one  day  the  whole 
company  take  9  pounds  of  gold  dust,  and  the  second  day 
take  as  many  pounds  as  those  new  members  number.  On 
the  third  day,  "  prospecting,"  the  whole  party  take  no  gold, 
but  lose,  each  man,  by  thieves  T|^  th  as  many  pounds  of  gold 
as  those  persons  numbered  who  last  joined  them,  when  the 
company  conclude  to  settle  up,  to  allow  the  members  to 
labor  individually.  It  is  required  to  adjust  the  number  of 
those  latest  members  to  the  greatest  luck  or  good  fortune 
of  an  individual  of  the  whole  for  the  three  days,  and  to 
determine  how  much  gold  was  a  share. 

Let  90  =  £,  9  =  a,  and  T^¥  =  c, 

and  x  •=.  the  new  members  ; 

then  one  member's  share  is 

a  -{-  x 

_  _,_.  j    st    rp   - •    nj 

dy  b—a 

—  =  — C  =  0  in  case  of  a  max.  or  mm., 

d  x       (b  -\-  x) 2 

.:X  =  ±L  V  ——  —  £  =  18  or  —  198, 


which  is  negative  under  the  values  which  these  constants 
are  known  to  have ;  indicating  a  maximum  for  y  while  x  is 
positive  at  18,  and  a  minimum  for  y  when  x  is  made  —  198. 
8 


86  DIFFERENTIAL   CALCULUS. 

However,  x  with  the  value  —  198,  has  no  application  to 
the  language  of  the  problem  as  enunciated. 

116,  This  occasion  is  taken  to  remark  that  differential 
coefficients  are  not  always  to  be  necessarily  regarded  as 
mere  numerical  amounts  or  ratios.  They  are  functions  as 
well,  and  it  is  useful  to  read  them  in  the  language  of  a 
problem's  special  kinds  of  quantity.  Thus,  in  regard  to  the 
problem  of  the  90  California  miners  as  first  stated,  the  first 
dif.  coef.  of  such  function  as  always  expresses  one  actual 
laborer's  share  of  gold  dust,  for  every  possible  number  of 
new  members,  which  coef.  is 

d  y  b  —  a 


x         (b  +  x)2 

when  read  as  a  derived  function  with  the  significance  of 
the  quantities  preserved,  may  be  as  it  follows  after  this 
preamble: 

One  actual  laborer's  share  in  pounds  will  always  be  found 
to  vary  (as  depending  on  the  variation  of  the  number  of 
new  men)  just"  as  the  following  supposed  share,  compared 
with  1  pound,  will  vary,  viz.  : 

From  as  many  pounds  of  gold  dust  as  the  original  com 
pany  numbered  men  (90),  take  what  the  workers  obtained 
the  second  day  (9),  and  divide  the  remainder  among  the 
actual  workers  as  their  number  would  be  after  each  worker 
had  withdrawn,  and  put  in  his  own  place  a  company  equal 
to  the  whole  workers,  and  then  take  away  from  each  such 
share  T|¥th  part  of  a  pound. 

28.  (a.)  A  and  B  set  out  at  the  same  time,  from  places 
320  miles  apart,  and  travel  to  meet.  Each  travels  uniformly 
at  his  own  rate,  and  the  number  of  hours  at  the  end  of  which 
they  meet,  is  equal  to  one  half  the  number  of  miles  which 
J?  goes  per  hour.  May  there  be  any  number  of  miles 
which  B  may  go  per  hour,  according  to  which  any  possible 


PROBLEMS   IN   MAXIMA    AND   MINIMA.  87 

difference  between  their  rates  per  hour  may  be  a  maximum 
or  minimum  ? 

(b.)  Less  than  at  what  rate  per  hour  can  they  not,  go, 
when  they  both  travel  alike  ?  If  they  travel  at  the  same 
rate,  what  is  that  ?  How  much  slower  may  A  go  than  _/?? 
B  than  A  ?  Ans.  Infinitely. 

(c.)  What  is  the  rate  of  each,  if  they  differ  3  miles  per 
hour  ? 

Ans.  Ambiguous,  because  it  is  not  hypothecated  which 
goes  the  faster. 

(d.)  Required  the  rate  of  each  when  A  goes  3  miles  per 
hour  faster  than  13,  and  the  rate  when  B  goes  3  miles 
faster  than  A. 

29.  Two  straight  lines,  A  B  and  A  C,  of  indefinite 
length,  meet  at  the  point  A  at  right  angles.  It  is  required 
to  determine  the  length  of  the  shortest  hypothenuse  that 
shall  pass  through  a  given  point  situated  in  the  plane  of 
those  two  lines,  at  the  perpendicular  distance  a  from  A  B, 
and  b  from  A  (7,  and  complete  a  right  angled  triangle. 

Let  x  =  that  part  of  A  C  not  equivalent  to  a  ; 
then  —  =  that  part  of  A  B  not  equivalent  to  5,  because 

*  ab 

x:b:a:  — ; 

X 

let  y  =  the  hypothenuse, 


then  y  =  V  («  +  ^)2  ±    *  + 


dy 2a62         2a2i2 

d~x  a  x2  x3     ' 

is  a  minimum  when  y1  is. 


88  DIFFERENTIAL   CALCULUS. 

This  problem  is  more  general  in  case  the  angle  at  A  is 
any,  and  the  hypothenuse  is  called  the  unspecified  side, 
but  requires  trigonometry. 

30.  Required    the    area   of  the   greatest   right-angled 
triangle  which  has  11  inches  for  its  hypothenuse ;  also, 
which  has  a  for  its  hypothenuse. 

31.  There  is  a  cylindric  tin  pail  without  a  cover,  of  which 
the  bottom  is  9  inches  in  diameter,  and  height  8  inches. 
Required  to  know  if  another  pail  can  be  made  that  may 
hold  as  much  water  with  less  sheet  tin,  and  how  much  loss ; 
and  required  the  rule  of  proportion  between  these  dimen 
sions,  both  when  having  a  flat  cover  and  when  without  one. 

32.  What  decimal  fraction  exceeds  its  cube  more  than 
any  other  numerical  quantity  whatever  exceeds  its  cube  ? 

Ans.  .577  +. 

33.  Divide  25  into  two  such  parts,  that  the  product  of 
the  second  power  of  one  part  by  the  third  power  of  the 
other,  may  be  larger  than  any  other  product  of  its  parts  at 
those  powers. 

Ans.  15  and  10,  the  larger  to  be  of  the  third  power. 

34.  (#.)  Two  farmers,  A  and  J?,  laid  out  for  themselves 
each  a  farm  of  equal  territory  and  rectangular  shape ;  a 
straight  line  drawn  from  a  corner  of  As  farm  across  it,  and 
meeting  a  side  68  rods  from  that  corner  which  is  diagonally 
opposite  the  corner  of  starting,  is  152  rods  long.     The  re 
mainder  of  the  side,  a  part  of  which  is  the  68  rods,  is  of 
the  same  length  as  one  side  of  ^Z?'s  farm.     Required  the 
length  and  breadth  of  each  farm  when  they  are  the  largest 
they  can  be  upon  these  conditions. 

(b.)  If  j5's  farm,  by  any  dimensions  we  may  adopt  for 
it,  is  square,  required  the  length  of  one  side ;  and  the  two 
sides  of  ^4's. 

35.  A  farmer,  having  at  first  80  dollars,  sold  3  times  as 
many  bushels  of  potatoes,  as  he  sold  them  at  in  cents  per 


PROBLEMS   IN   MAXIMA   AND   MINIMA.  89 

bushel,  and  then  purchased  246  bushels  ot  corn,  each 
bushel  at  the  price  of  a  bushel  of  potatoes  just  sold,  and 
has  left  a  least  possible  or  a  greatest  possible  amount  of 
money.  Required  whether  greatest  or  least,  and  the  prices 
of  the  potatoes  and  corn  per  bushel. 

36.  From  the  equator  a  ship  sailed  north  50  times  as 
many  hours  as  she  sailed  miles  per  hour ;  thence  she  sailed 
south  1000  hours  at  the  same  rate,  and  the  result  was  the 
least   possible   gain   to  the   north.     Required  how  many 
miles  per  hour  she  sailed,  and  how  far  she  is  from  the 
equator. 

37.  (a.)  Supposing  the  Boston  and  Worcester  Railroad 
and  the  Old  Colony  Railroad  to  run  from  Boston  at  right 
angles  to  each  other,  and  the  roads  straight,  and  that  a 
train  of  cars  on  the  Worcester  road,  19  miles  from  Boston, 
is  ready  to  start,  headed  for  Boston,  at  32  miles  an  hour, 
and  a  train  on  the  Old  Colony  is  ready  to  leave  Boston  at 
the  same  instant,  at  20  miles  an  hour,  how  far  from  Boston 
will  each  train  be  when  they  are  nearest  together  by  a 
straight  line  across  the  country,  and  how  long  after  starting, 
and  how  far  apart  then  ? 

Ans.  In  part,  the  train  on  the  Worcester  road  5£f£ 
miles  from  Boston. 

(&.)  Repeat  the  problem,  with  the  conditions  all  the  same, 
except  that  the  Old  Colony  train  is  to  have  25  miles  the 
start  of  the  former  condition,  and  see  if  any  indication  is 
offered  that  the  occurrence  we  are  watching  for  must  have 
already  happened,  or  would  occur  in  the  future,  on  each 
train  reversing  its  direction,  and  where  will  the  condition 
exist,  and  how  long  after  starting  from  this  position. 

38.  It  is  required  to  find  a  numerical  quantity  such  that 
if  from  9  times  itself  its  second  power  be  subtracted,  the 
remainder  will  be  equal  to  3  times  the  quantity  plus  another 
sum  ;  what  is  the   quantity  when  this   other  sum  is  the 
greatest  or  least  possible,  and  which  of  the  two  ? 

8* 


90  DIFFERENTIAL  CALCULUS. 

39.  The  sum  of  two  quantities  is  22  ;  the  second  power 
of  one  added  to  twice  the  second  power  of  the  other,  is  a 
maximum  or  minimum  ;  which  ?     Required  the  numbers. 

40.  The  difference  between  two  quantities  is  10,  and  the 
difference  of  their  third  powers  is  a  maximum  or  minimum. 
What  are  they  ? 

41.  A  wholesale  druggist  bought  542  (a)  pounds  of  a 
drug  at  $3.57  (b)  per  pound,  and  sold  from  it,  a  part  at  the 
same  number  of  cents  per  pound  as  equals  the   number 
of  pounds  not  sold,  and  the  whole  amount  of  the  profit  or 
loss  on  this  sale  was  the  greatest  for  any  quantity  sold  ; 
at  what  price  was  that  portion  sold  per  pound,  and  what 
the  amount  of  the  profit  or  loss  on  that  portion  as  a  whole  ? 

Let  x  =  cents  per  pound  of  that  sold  ; 

.-.  x  =  the  number  of  pounds  not  sold  ; 
.•.a  —  x  =  the  pounds  sold  ; 

.*.  ±  x  ^f  b  •=.  profit  or  loss  per  pound  on  that  sold. 
Let     y  =  (a  —  x)  (±  x  ^p  b)  =  all  the  profit  or  loss  ; 
'.  y,  i.  e.,  ±  (a  +  b)  x  +  #2  =f  a  b  =  max.  or  min., 


.-.  y  is  a  maximum  for  profit,  and  minimum  for  loss,  at  the 
same  value  of  x. 

Since  x  ^>  5,  the  sale  is  known  to  be  at  a  profit,  /.  y  = 
max.,  because  the  negative  sign  of  second  dif.  coef.  be 
comes  adopted. 

NOTE.  In  the  URC  of  the  doubtful  sign  ±,  care  should  be  taken  to  preserve 
throughout  the  above  work  their  proper  correlation  ;  the  upper  all  belonging 
together  in  the  logical  relation,  or  the  under. 


PROBLEMS   IN   MAXIMA   AND   MINIMA.  91 

42.  In  a  certain  country  may  be  found  a  factory  which 
turns  out  toys.     Factory  No.  1  makes  a  toy,  which  it  packs 
in  a  paper;  and  with  regard  to  this  factory,  one  such  pack 
age  may  be  indifferently  called  a  paper,  package,  case,  or 
box.     Factory  No.  2  puts  2  of  its  toys  in  a  paper,  2  papers 
in  a  package,  2  packages  in  a  case,  2  cases  in  a  box.     Fac 
tory  No.  3  does  the  same,  except  it  puts  3  toys  in  a  paper, 
3  papers  in  a  package,  3  packages  in  a  case,  3  cases  in  a 
box.     Factories  No.  4,  5,  6,  etc.,  put  4,  5,  6,  etc.,  respec 
tively,  toys  up  by  the  same  rule  of  packing;  —  each  factory 
commencing  with  one  more  toy,  paper,  package,  and  case 
than  its  predecessor,  and  so  on  with  factories  indefinitely, 
Now,  in  one  of  the  two  packing  rooms  in  each  factory,  the 
last  stage  is  the  packed  case,  in  the  other,  the  last  stage  is 
the  packed  box.     Suppose  that  in  the  case  room  of  each 
factory,  there  are  144  packed  cases ;  enough  of  these  are 
carried  into  the  box  room  to  make  3  boxes,  if  possible. 
These  boxes  are  then  collected  and  shipped.     Remaining 
in  the  case  room  of  which  f-ictory,  is  there  the  greatest 
number  of  individual  toys  ?     In  which  remain  none  ?  and 
between  what  factories  is  the  difference  the  greatest,  in  the 
number  of  toys  left  in  the  case  rooms  ? 

Ans.  In  the  36th  is  the  greatest  number ;  in  the  48th 
none ;  the  difference  the  greatest  between  the  23d 
and  24th  ;  in  the  23d,  toys  912,525  ;  in  the  24th, 
995,328  ;  in  the  25th,  1,077,375  ;  difference  between 
23d  and  24th,  82,803  ;  24th  and  25th,  82,047. 

It  is  obvious  that  the  problem  cannot  have  practicability 
under  such  generalizations  of  the  constants,  as  would  ren 
der  the  above  results  fractional.  Thus,  if  144  were  to  be 
replaced  with  141,  the  36th  in  the  above  answer  becomes 
35  f 

43.  The  sides  for  constructing  a  quadrilateral  plane  fig 
ure,  are   consecutively  27,  19,  42,  and  31  units  of  length, 


92  DIFFERENTIAL   CALCULUS. 

any  three  of  which  are  greater  than  the  fourth  ;  it  is 
required  to  determine  the  diagonal  meeting  the  sides  19  and 
42,  and  that  meeting  the  sides  42  and  31,  when  the  area  of 
the  quadrilateral  is  the  greatest  possible.  Required  that 
area  in  square  units  of  the  same  name. 

44.  A  man  having  a  sum  of  money,  gave  90  dollars  of 
it  to  a  charity  fund,  and  distributed  the  remainder  of  it  to 
the  same  number  of  deserving  orphans,  as  dollars  to  each, 
one  of  whom  was  T.  Now,  there  were  some  schools  close 
by,  as  many  in  number  as  T  received  dollars,  each  school 
having  as  many  boys,  and  each  boy  owning  as  many  dol 
lars  as  T  received,  or  owing  as  many,  for  we  did  not  hear 
distinctly,  but  consider  that  it  might  have  been  either  way. 
Required  to  determine  the  relation  between  the  sum  of 
dollars  originally  possessed  by  the  man,  and  the  possessions 
in  dollars,  or  debts  of  these  boys,  and  their  relative  rates 
of  variation  for  all  possible  sums. 

Let  x  =  the  man's  sum  ; 

and  y  •=.  all  the  boys'  sum  ; 

.:y  =  ±  (x  —  90)1; 


Here  —  may  =  0,  but   y  has   no   maximum   or   mini- 

d  x 

mum,  because  d  F  (a;  —  h)  is  imaginary  when  x  =.  90. 

45.  A  man  having  a  number  of  dollars  in  a  purse,  put 
10  dollars  of  it  into  a  Savings  Bank,  and  having  divided 
the  remainder  into  as  many  parts  as  he  put  dollars  in  a 
part,  proceeded,  having  other  money  in  a  pocket  book,  to 
purchase  some  articles,  which  were  just  22  in  number  less 
than  the  original  number  of  dollars  in  the  purse,  paying  for 
each  article  a  sum  equal  to  one  of  those  parts.  Required 


PROBLEMS   IN   MAXIMA   AND   MINIMA.  93 

to  trace  all  the  relations  between  the  number  of  dollars  in 
that  purse,  and  the  value  of  all  those  articles  purchased. 


The  function  (x  —  22)  *J  x  —  10  =  y,  has  the  value 
zero  when  x  =  22,  and  when  x  =  10,  and  an  algebraic 
maximum  and  minimum  when  x  =  16,  which  value  is  in 
compatible  with  the  conditions.  After  x  exceeds  22,  y  is 
practicable  to  infinity  in  its  positive  values. 


DEFINITIONS. 

117.  A.  pyramid  is  a  solid  figure  contained  by  planes, 
that  are  constituted  between  one  plane  called  the  base,  and 
a  point  above  it  called  the  apex,  in  which  they  meet. 

A  prism  is  a  solid  figure  contained  by  plane  figures,  of 
which  two  that  are  opposite  are  equal,  similar,  and  parallel 
to  one  another ;  and  the  others  are  parallelograms. 

A  parallelepiped  is  a  solid  figure  contained  by  six  quadri 
lateral  figures,  whereof  every  opposite  two  are  parallel. 

A  right  pyramid  has  its  apex  perpendicularly  over  the 
centre  of  its  base,  when  that  base  has  such  regularity  as  to 
have  a  centre,  equally  distant  from  the  termination  of  each 
side  of  that  base. 

A  right  prism  has  rectangles  for  such  of  its  sides  as  must 
be  parallelograms. 

A  right  parallelepiped  is  contained  by  no  other  plane 
figures  but  rectangles. 

46.  (a.)  The  base  of  a  right  pyramid  is  triangular,  of 
which  the  sides  are  «,  5,  and  c,  and  height  is  II\  it  is 
required  to  find  the  contents  of  the  largest  right  prism 


94  DIFFERENTIAL   CALCULUS. 

which  can  be  contained  in  it,  and  each  linear  outline  dimen 
sion  of  the  same. 

Ans.  Contents,  —  II X  -  X  area  of  base. 

3  9 

2a      2b      2c  1     . 

Linear  outline, — ,   — ,   — ,  and  -  H. 
333  3 

(b.)  The  base  of  a  pyramid  is  triangular,  of  which  the 
sides  are  a,  £,  and  c,  and  the  perpendicular  height  is  H\  it 
is  required  to  find  the  sides  of  the  triangles  that  are  par 
allel,  containing  the  largest  prism  that  can  be  contained  in 
the  pyramid. 

(c.)  The  base  of  a  right  pyramid  is  rectangular  a  by  #, 
and  perpendicular  height  is  II\  required  each  linear  out 
line  dimension  of  the  largest  parallelepiped  that  can  be 
contained  in  it. 

(d.)  The  base  of  a  pyramid  is  quadrilateral,  the  sides 
being  a,  #,  c,  and  /;  the  area  of  the  base  A,  and  perpen 
dicular  height  is  H.  Of  such  two  like  plane  figures  as  are 
parallel  to  each  other,  and  are  determinate,  and  bound  the 
largest  prism  that  can  be  contained  in  the  pyramid,  required 
the  sides.  Required  also  the  ratio  of  the  contents  of  the 
prism  to  those  of  the  pyramid.  Ans.  Ratio  $-ths. 

47.  A  hound  discovers  a  deer  1800  (a)  feet  ahead  of 
him  :  they  both  start  the  same  instant,  the  deer  in  a  direct 
line,  pursued  by  the  hound  ;  the  hound  makes  a  leap  each 
second,  and  1  foot  longer  than  the  deer's,  and  makes  3  more 
leaps  than  the  deer  does  in  the  time  that  the  deer  makes 
one  less  leaps  than  the  number  of  feet  in  its  length. 

After  taking  120  (b)  leaps,  the  deer  becomes  arrested  by 
the  breaking  of  the  crust  of  snow ;  at  this  instant  the 
hound  seeing  the  deer  arrested,  adds  2  feet  to  the  length 
of  his  previous  leap,  and  makes  7  in  the  time  that  he  made 
6  before,  which  he  continues  till  he  comes  up  with  the  deer. 
Now,  if  the  deer  was  occupied  in  extricating  himself,  or  in 
resting  during  the  greatest  or  least  time  possible,  how 


PROBLEMS  IN  MAXIMA   AND  MINIMA.  95 

long  had  been  the  deer's  leap,  to  accommodate  this  con 
dition  ? 

Let     x  =  feet  leaped  by  the  deer  at  a  leap, 
and         y  =  number  of  leaps  of  hound  after  deer  stops  ; 

/.  b  x  =  all  the  feet  leaped  by  the  deer  ; 
/.  x  -\-  1  =  feet  of  hound's  leap  at  first  ; 

x  +  2 
/.  --  X  ft  (a  +  1)  =  h.'s  dis.  attained  when  d.  stops  ; 

/.  a  -|-  b  x  =  whole  distance  for  hound  to  go  ; 
a-\-bx  ---  (  b  x  -\-  b)  =  to  be  leaped  by  h.  after  d.  stops  ; 

let 

let  e  =  a  —  4  b  ; 

and  g  —  -f-  a  -f  2  b  ; 


c  (x  +  3) 


let  y'= 


ex  —  g 


-*-'  —  ex2  —  3e  +  2ff         At, 

•  •  —  = =  0,  when  y  =  max.  or  mm. ; 

(L  y  ( r%  — i-u  9  T         r\\%  ** 

/.  the  numerator  =  0  ; 


2^~  3  e  +      =  12.66,  or —  9.46; 


96  DIFFERENTIAL   CALCULUS. 

d*y>         QE24-2a;  —  3)(2g  —  2  ear)  —  (2x  +  2)(2gx  —  ex*  — 


dx*  (a;2  +  2  a;  —  3)4 

6 

now  y  =  y'  X  -• 

d2  y' 

Now,  y  is  a  max.  when  y  '  is,  as  is  evident,  and  -    -  is 


negative  when  x  =  12.66  ;  hence  a  maximum  which  is  an 
answer,  and  the  negative  value  of  x,  is  not  within  the 
significance  of  the  language  of  the  problem. 


SECTION    XIII. 
COMPLETE  HISTORY   OF  FUNCTIONS. 

118.  It  is  useful  to  become  acquainted  with  the  methods 
of  fully  examining  the  entire  history  of  a  function  of  one 
or  more  variables,  in  respect  to  the  range  of  values  which 
the  function  and  its  variable  may  sustain,  and  to  their 
mutual  dependence.  Attention  should  also  be  paid  to  every 
constant  in  its  influence  on  the  function.  All  the  promi 
nent  characteristics  of  value,  and  the  rates  of  change  of 
value,  should  be  noted,  by  special  regard  to  the  value  of 
dif.  coefs. 

A  function  should  be  tested  for  the  values  -|-  o>  and 
—  co,  and  for  how  many  times  or  successions  it  has  either, 
and  at  what  values  of  the  variable  respectively ;  whether 
it  passes  from  -)-  co  to  —  co,  or  from  —  co  to  -\-  co  with 
instanitaneity,  so  to  speak,  as  when  we  should  examine 
y  =  x~l  and  2/  =  ie~2,  and  observe  the  marked  differ 
ence  in  respect  to  x  ^>  0  or  x  <^  0,  i.  e.,  while  x  passes  the 
value  0. 

A  function  should  be  tested  for  what  it  becomes  when 


COMPLETE   HISTORY   OF   FUNCTIONS.  97 

the  variable  =  0,  and  for  whether  it  has  more  than  one 
value  for  the  variable  =  0  ;  it  should  be  tested  for  the 
variable  —  -f-  co  and  =  —  c/>. 

A  function  should  be  tested  for  its  value,  zero,  for  how 
many  times  it  has  such,  if  at  all ;  for  at  what  value  or 
values  of  the  variable  it  has  such,  whether  it  passes  through 
such  value  or  not,  and  if  through,  at  what  rate  of  change. 

A  function  should  be  tested  for  maxima  and  minima, 
for  how  many  times  it  has  either,  and  at  what  values  of  the 
variable ;  the  same  dif.  coefs.  may  be  employed  to  deter 
mine  with  facility,  whether  the  variable  of  itself  has  maxima 
or  minima  ;  and  if  so,  at  what  values  of  the  function. 

Since  the  constants  that  occur  in  a  function  are  quan 
tities,  with  which  both  the  variable  and  the  function  are 
most  concerned,  or  with  which  they  are  most  compared, 
and  in  consequence  are  liable  to  exhibit  marked  character 
istics  when  they  become  equal  to  or  pass  the  value  of  such 
constants,  tests  should  be  applied  for  the  values  of  both 
the  variable  and  the  function,  when  either  is  equal  to  such 
constants,  but  sometimes  to  a  combination  of  such  con 
stants,  as  a  product,  sum,  or  quotient. 

Functions  of  a  single  variable  should  be  studied  with 
reference  to  interrupted  values,  i.  e.,  to  losses  of  con 
tinuity. 

A  function  should  be  tested  for  the  maximum  or  minimum 
of  its  rates  of  change  of  value. 

A  function  should  be  tested  for  proof  of  symmetrical- 
ness,  or  having  the  same  value  when  its  variable  is  a  given 
amount  less  and  more  than  some  specific  quantity  or  0 ; 
and  the  variable  be  tested  for  having  the  same  value  when 
the  function  is  a  certain  amount  less  and  more  than  a 
specific  quantity  or  0. 

It  will  be  necessary  to  be  prepared  for  cases  in  which 
neither  the  function  nor  variable  can  have  any  real  values 
whatsoever. 

9 


98  DIFFERENTIAL   CALCULUS. 

The  nature  of  these  inquiries  may  be  shown  by  an 
instance : 

1.  Let  it  be  required  to  determine  the  range  of  all  pos 
sible  parts  into  which  the  number  50  might  be  divided, 
both  when  one  of  the  parts  is  not  greater  than  50,  and 
when  it  is  any  number  whatsoever,  estimated  by  algebraical 
equivalents ;  or, 

Required  the  chief  points  in  the  history  of  the  function, 

y  or  F  x  =  50  x  —  #2  =  (50  —  x)  x. 

F  x  will  be  found  to  have  a  maximum  when,  and  only 
when,  x  =  25,  F  x  being  625,  for  there  is  but  one  value  of 
x  at  which  there  is  any  maximum.     It  can  have  no  mini- 
d2  y 

mum,  because  — g  is  always  —  2,  and  of  course  such  when 

a  =  25. 

F  x  cannot  be  so  great  as  625  at  any  other  value  of  x 
than  25,  because  in  the  equation, 

50  x  —  x*  =  625, 

x  will  be  found  to  have  but  one  value,  viz.,  x  =  25  ;  F  x, 
therefore,  cannot  =  -f-  co  in  value. 

F  x  may  =  —  c/>  both  when  x  =  —  co,  and  when  x  — 
-f-  co,  for,  solve  the  equation, 

50  x  —  x*  =  —  co, 


.-.  x  =  25  ±  V  *>  +  625  =  ±  co. 

F  x  has  the  value  0,  both  when  x  =  0  and  when  x  = 
50,  for,  solve  the  equation, 

50  x  —  x*  =  0, 
.-.  x  =  0  or  50. 
Such  are  the  general  outlines  of  the  history  of  possible 


COMPLETE   HISTORY   OF   FUNCTIONS.  99 

values  of  F  x  and  of  x.  Commencing  now,  with  x  alge 
braically  the  smallest  conceivable,  or  =  —  eo,  F  x  being 
also  —  eo,  we  have, 

y  •==.  50  x  —  x^  =  —  eo, 
and  x  =  —  eo 


=+«; 

because  —  —50  —  2^  =  50  —  (2  X  —  co)=50+co=eo. 

d  x 

Since  the  first  dif.  coef.  =  eo  when  x  =  —  eo,  F  x  is 

indicated  about  to  increase  or  become  a  smaller  negative, 

d2  y 
at  an  infinite  positive  rate.     And  since  —  is  always  —  2, 

such  rate  of  increase  is  ever  to  diminish,  and  we  know  that 
50  —  2  x  grows  less  as  x  increases.  These  conditions  con 
tinue  till  F  x  =  0  and  x  =  0,  when  —  =  50  ;  hence  F  x 

d  x 

passes  through  zero,  increasing  50  times  as  fast  as  #,  and 
beginning  now  to  have  positive  values,  goes  on  till  F  x  = 

625,  and  x  =  25,  when  —  =  0.     F  x  now  returns  to  have 
d  x 

a  less  value,  till  it  =  0,  x  then  being  50  ;  in  diminishing, 
F  x  passes  through  the  value  0,  while 

—  =  50  —  2x  =  50  —  2X50:=:  —  50; 

d  x 

which  indicates  that  F  x  passes  again  through  zero,  dimin 
ishing  50  times  as  fast  as  x  :  so  F  x  goes  on  to  =  —  eo, 
already  shown,  when  x  =  4-  eo  and  —  =  —  eo,  as  shown. 

d  x 

Again,  F  x  is  symmetrical  in  its  history  before  and  after 
its  maximum  ;  i.  e.,  while  x  is  any  amount  greater  or  less 


100  DIFFERENTIAL   CALCULUS. 

than  25,  F  x  has  the  same  value.     This  may  be  verified 
thus  : 

Let  x  =  v  -\-  25, 

and  x  =  25  —  w  ; 

=  50(25  —  w)  —  (25  —  w)2, 


whenever  w  =  v,  because,  on  such  condition,  the  members 
of  the  equation  reduce  to  identity. 

If  the  given  function  should  be  as  general  as 

a  xn  —  b  x*n, 
this  characteristic  of  symmetricalness  could  be  shown. 

2.  Let  it  be  required  to  determine  whether  the  equation 
of  the  Second  Degree  between  two  variables,  x  and  y,  viz.  : 

50  x  —  x*  —  y  =  0,  t 

involves  the  elliptic,  hyperbolic,  or  parabolic  condition. 
(Arts.  50,  51,  52,)  An's.  The  parabolic. 

3.  It  is  required  to  determine  whether  F  x  —  —  x3  -\- 
3  a;2  -f-  24  x  —  85,  has  a  maximum  and  a  minimum  at  any 
values  of  a?,  and  to  explain  how  it  can  be  -|-  co  and  —  c/> 
also. 

4.  It  is  required  from  F  y  =  x,  viz.,  50  y  —  y  2  =  x,  to 
obtain  some  function  of  x  equal  to  y,  and  recount  its 
history. 

5.  Required  the  complete  history  of  y  =  —  . 

6.  Required  the  complete  history  of  y  =  -  . 

7.  Required  the  complete  history  of  y  =  -  . 

b  —  #8 


PROJECTED   BODIES.  101 


SECTION  XIV. 


PRINCIPLES  AND  PROBLEMS  RELATING  TO  PROJECTED 
BODIES. 

119.  The  height  of  any  point  in  the  course  which  a 
dense  body  like  a  stone  or  a  mass  of  water  takes,  when 
thrown  in  any  direction  whatever,  near  the  earth  and 
through  a  medium  no  denser  than  the  air,  and  at  velocities 
not  exceeding  300  or  400  feet  per  second,  may  be  almost 
exactly  expressed  by  this  function  of  its  attained  horizontal 
distance  at  any  the  same  point,  x  being  that  distance,  viz.: 


=  »(— 7> 


in  which  a  represents  the  whole  horizontal  distance  attained, 
with  reference  to  the  level  of  the  point  of  commencing  to 
.move,  or  rather  of  the  commencing  calculation,  and  in 
which  b  represents  such  a  number  of  times,  or  is  such  a 
factor  to  one  fourth  of  «,  as  equals  or  expresses  the  greatest 
height  attained  by  the  body  if  uninterrupted.  But  b  may 
be  fractional. 

The  demonstration  of  this  formula  belongs  to  Mechanics. 

130.  In  the  particular  case  when  the  body  is  thrown 
most  favorably  to  attain  the  greatest  range  or  horizontal 
distance  for  the  force  used,  b  becomes  of  the  value  1,  and 

may  consequently  be  erased  from  the  formula.  For  —  may 
be  found  to  be  the  maximum  value  of 


and  this  value  accrues  to  the  function  y,  when  x  =  — 


102  DIFFERENTIAL   CALCULUS. 

131,*  Iri  the  particular  case,  when  the  body  is  thrown 
perp  endi-eularly  upward,  since  x  =  0  and  a  =  0,  the  gen 
eral  formula, 


being  restored,  it  would,  at  first  thought,  appear  that  y,  or 
the  height  attained,  must  be  zero,  which  would  be  absurd  ; 
but  b  being  conditioned  to  be  equal  to  such  a  number  of 

times  -  (=  0)  as  would  make  the  actual  greatest  height 
attained,  must  be  infinite,  so  that 

co  X  0  =  y, 
where  y  may  still  have  any  finite  value. 


In  the  particular  case,  when  the  body  is  thrown 
horizontally,  and  attains  no  height  after  starting,  each  of  a 
and  b  are  zero,  and  the  one  only  possible  value  of  x  is  zero, 
with  reference  to  that  line  of  level.  But  a  new  line  of  level 
may  be  any  where  assumed,  or  the  point  where  the  body 
strikes  or  finds  its  course  interrupted.  Indeed,  the  descent 
of  a  thrown  body  is  in  its  course,  with  reference  to  horizon- 
tality,  symmetrical  with  its  ascent,  and  we  have  shown 
elsewhere  the  symmetricalness  of  the  function 

50  x  —  x2  =  y, 

V  X^ 

which,  if  50  =  a  and  —  =  y',  becomes  x  --  =  y'\  but 

it  may  have  any  factor  or  #,  and  the  symmetricalness 
shown.  Hence,  the  point  of  interrupted  motion  of  a  thrown 
body  may  be  considered,  for  all  the  purposes  of  these  cal 
culations,  as  the  point  of  projection. 


Hence,  a  body  thrown  descendingly,  describes  the 
course  of  a  body  thrown  upwardly,  and  having  its  upward 


PROJECTED   BODIES.  103 

motion  interrupted  before  it  may  be  completed  ;  this  being 
so,  whether  directly  downward  or  laterally  be  the  direction. 

124.  And  hence  further,  a  body  allowed  to  fall  directly 
down,  is  thrown  by  its  gravity  ;  and  its  point  of  setting  out 
is  the  same  as  the  point  of  greatest  height  that  would  be 
attained  in  the  case  of  the  same  body  thrown  directly  up, 
with  the  force  such  as  it  would  have  acquired  in  a  fall 
through  that  same  distance. 

125,  If  the  function  given  as  expressing  the  variable 
height  of  the  thrown  body,  be  differentiated,  we  have 


dy  2b 

-  =b  --  x. 

ax  a 

The  value  of  this  differential  coefficient  expresses  the 
direction  of  motion  of  the  body,  while  at  the  distance  x 
on  its  course,  with  reference  to  horizontally  and  perpen 
dicularity  ;  it  is  their  ratio,  d  y  being  upward,  d  x  lateral. 
This  coefficient  is  evidently  greatest  in  practicalness  when 
x  =  0  ;  therefore  a  body  is  making  upward  the  most  direct 
ly  at  its  point  of  setting  out.  It  is  zero  when  x  =  £  a  ;  the 
body  is  then  for  an  instant  of  time  moving  horizontally  ; 
maximum  height  is  attained. 

In  the  case  when  a  body  is  thrown  in  the  most  favorable 
direction  to  attain  range  for  the  given  force,  since  b  =  1, 
this  coefficient  then  =  1,  consequently  the  body  is  thrown 
in  the  direction  of  the  hypothenuse  of  a  right-angled  tri 
angle,  such  as  has  its  base  and  height  equal,  or  at  an  angle 
of  45  degrees. 

126,  When  an  elastic  thrown  body  strikes  a  firm  per 
pendicular  plane,  as  when  a  playing  ball  strikes  the  side  of 
a  building,  the  course  and  distance  of  the  rebound  are  almost 


104  DIFFERENTIAL   CALCULUS. 

the  complement  of  the  course  and  distance  as  they  would 
have  been  had  there  been  no  interruption. 

127.  Since  the  points  of  the  setting  out  and  of  arrest 
of  a  thrown  body  in  practice,  are  determined  by  human 
economy,  or  the  presence  and  interruption  of  the  earth, 
such  points  are  special,  and  have  no  significance  in  the  gen 
eral  mathematical  indications  of  its  course,  which  are  more 
completely  fulfilled  in  the  case  of  moving  celestial  bodies. 
The  thrown  body  may  be  conceived  to  have  come  out  of 
the  earth,  and  to  again  pass  into  or  through  it. 

128.  If,  furthermore,  we  will   discharge  the  condition 
that  the  successive  lines  of  the  measured  heights  of  the 
thrown  body  are  to  be  parallel,  and  will  assume  them  to  be 
parts  of  the  radii  of  the  earth,  which  is  the  more  proper 
consideration,  then  will  the  thrown  body  return  in  its  orbit 
into  and  out  of  the  earth,  or  through  it  and  back,  and  con 
tinue  to  revolve  forever.     But  the  formula  would  need 
some  change  to  be  rendered  compatible  with  this  course. 
Hence,  further,  a  body  thrown  or  dropped  towards  the 
centre   of  the    earth's  gravity,   must,   after  going  as   far 
beyond,  return  in  the  same  line,  and  oscillate  forever,  if 
uninterrupted. 

129.  If  a  body  could  be  conceived  as  thrown  with  an 
infinite  force,  so  that  its  range,  a,  may  be  also  infinite,  the 
formula 


(x*\ 
x  --  )  =  y,  becomes  b  x  = 
a    ' 


y, 


because  the  fraction  of  which  a  is  the  denominator,  be 
comes  zero  (unless  x  is  also  infinite,  when  which  is  the 
case,  the  fraction  has  an  indeterminate  value).  The  equa 
tion  reduces  to  one  of  the  first  degree  ;  its  first  differential 
coefficient  is  always  b  or  constant  ;  hence  the  body  moves 
in  a  straight  line  forever. 


PROJECTED  BODIES.  105 

A  projected  body  always  moves  in  a  plane  that  is  per 
pendicular  to  the  horizon,  or  very  nearly  so. 

1.  (a.)  A  stone  is  so  thrown  as  to  reach  the  greatest 
distance,  and  its  greatest  height  attained  is  40  feet.     Re 
quired  the  distance  of  its  arrest  or  its  range. 

Ans.  160  feet. 

(b.)  Required  the  value  of  ratio  of  its  upward  and  for 
ward  directions  at  its  start.  ,  Ans.  1. 

(c.)  Required  the  same  after  it  had  attained  the  eleva 
tion  of  16  feet,  or  while  possessing  that  elevation. 

In  this  case, 

x  -   ~Q  =  16  .-.  x  =  18  or  142  ; 

d>y  __  2  X  18 31 

"  dx~        ~     160     '  ~  S)' 

^  __  2  X  142  _  31 

dx~  160  40 " 

The  former  of  these  results  is  adapted  to  the  ascension, 
the  latter  to  the  descension. 

2.  A  stone  is  so  thrown  as  to  attain  for  its  greatest 
height  62  feet,  and  distance  142  feet;  it  is  required  to  de 
termine  how  far  it  was  from,  when  directly  over  or  under, 
as  it  may  be,  the  telegraph  wire,  which  crosses  the  plane 
of  the  stone's  motion,  at  40  feet  distance,  horizontal,  and 
height  50  feet. 

8.  A  steam  fire-engine  threw  water,  the  pipe  being 
directed  to  a  point  30  feet  high  at  70  feet  distance ;  the 
force  used  carried  the  water  a  distance  of  274  feet,  meas 
ured  on  the  level  of  the  mouth  of  the  pipe.  Required  the 
greatest  height  attained  by  the  water  above  that  level. 

Ans.  29.35  feet. 


106  DIFFERENTIAL   CALCULUS. 

4.  From  an  elevation  of  44  feet  a  body  is  thrown  in  its 
first  outset  downwardly  8  and  laterally  5,  in  direction  of 
aim  ;  the  force  used  was  such  that  had  it  been  exerted  in 
precisely  the  opposite  direction,  the  body  would  have  risen 
to  17  feet  greater  height  than  that  elevation.     Required  to 
determine  how  far  from  the  foot  of  that  elevation  the  body 
struck  the  ground.     The  direction  of  aim  is  supposed  to  be 
tangent  to  the  course  adopted  by  the  body. 

In  this  case  we  readily  find  the  range  the  body  would 
have  made  on  the  level  of  its  setting  out,  had  it  been 
thrown  in  precisely  the  opposite  direction,  to  be  108  feet  = 
a ;  then  with  the  variables  x  and  y  adapted  to  this  level, 

we  find  —  for  the  value  of  y  =  —  44,  which  value  we  will 
denote  as,  dy,  ^  ^  M2 

d~x<  ~  "  135' 

o 

to  obtain  which,  b  =  -,  was  called  negative.     Adopting 
5 

now  the  lower  level  as  the  basis  of  calculation,  x'  and  y' 
as  the  variables,  we  have  17  -\-  44,  to  find  a',  the  lower 
range,  from  one  half  of  which  we  subtract  one  half  of  a 

cl  *u 

for  the  answer ;  the  negative  sign  of  —  is  adapted  to  de- 

d  x 

scent,  and  may  be  called  positive  with  reference  to  the 
result  desired,  if  the  motion  of  the  thrown  body  be  con 
strued  as  upward. 

5.  A  person  undertaking  to  draw  a  liquid  from  a  small 
hole  bored  horizontally  into  the  head  of  a  barrel,  observed 
that  when  the  stream  had  obtained  the  downward  descent 
of  7  inches,  its  lateral  reach  was  9  inches  from  the  barrel's 
head,  the  same  being  supposed  perpendicular.     The  hole 
was  27  inches  above  the  floor,  upon  which  stood  a  cylindric 
tin  vessel  of  4  inches  diameter  at  the  top  and  of  6  inches 
depth ;  the  central  axis  of  the  vessel  in  the  plane  of  the 


PROJECTED   BODIES.  107 

stream  was  18  inches  laterally  away  from  the  perpendicular 
plane  of  the  barrel's  head.  Required  to  determine  whether 
the  liquid  was  caught ;  the  diameter  of  the  stream  being 
considered  zero. 

130.  It  is  an  important  general  principle  to  state,  that 
the  direction  of  the  aim  of  an  upwardly  thrown  body  is 
always  towards  a  point  just  twice  as  high  as  the  body  ever 
reaches  at  its  greatest  height,  when  such  point  is  taken 
perpendicularly  above  the  point  of  the  greatest  elevation 
that  is  to  be  actually  attained  without  interruption.  Hence, 
in  the  problem  of  the  drawn  liquid,  the  differential  coeffi 
cient  of  the  function  descriptive  of  the  course  of  the  stream 

at  the  point  7  inches  down,  9  inches  lateral  change,  is  - , 

9 

if  the  direction  of  motion  be  considered  reversed.  But 
the  most  direct  way  of  solving  this  problem  is  on  the 
principle,  true  with  reference  to  all  falling  bodies,  when 
the  point  of  commencing  descent  is  given,  that  distances 
attained  downward  from  this  point  are  direct  to  each  other 
as  the  squares  of  the  laterally  attained  distance ;  hence, 


Z  being  the  lateral  distance  of  stream  when  it  is  down  21 

inches. 

131.  If  the  formula  (Art.  119),  be  considered  with  refer 
ence  to  the  Degree  of  its  Equation,  it  will  be  found  to  be 
of  the  Second,  and  to  involve  the  Parabolic  condition  (48), 
because  in 


i.  e.,  a  y  -f-  a  b  x  —  b  x*  =  0, 

we  have  a  =  D,  ab  =  E,  b  =  <7,  0  =  B,  and  0  =  A ; 
...  j?2  _  4  A  c  =  0. 


108  DIFFERENTIAL   CALCULUS. 

The  change  of  the  formula  suggested  in  Art.  138,  but 
omitted,  would  give  us  an  equation  involving  the  Elliptical 
condition. 

Gunshot  projectiles  being  thrown  at  velocities  varying 
from  900  to  2000  feet  per  second,  the  resistance  of  the  air 
condensed  before  the  moving  body  becomes  so  great  that 
the  formula  given  at  (119)  is  unavailable ;  the  body  falls 
short  of  the  distance  indicated  by  the  law  of  that  formula. 


SECTION  XV. 

THE  SYSTEM  OF  SYMBOLS  FOR  FUNCTIONS.  —  AN  IM 
PLICIT  FUNCTION  OF  A  SINGLE  INDEPENDENT  VARI 
ABLE  AND  ITS  DIFFERENTIATION. 


We  have  shown  the  nature  and  purpose  of  a  func 
tion  of  a  single  independent  variable  when  it  is  explicit, 
and  have  exhibited  the  method  of  notation  of  it  as  (a  func 
tion  of  x)  =  y.  When  more  than  (one  function  of  x)  =  y 
are  employed  in  one  investigation,  it  will  be  necessary  to 
discriminate  them  ;  they  are  accordingly  discriminated,  as 
F  x,  f  x,  f  x,  f"  x,  etc.  Whenever  we  are  concerned 
with  two  functions  of  #,  which  need  not  be  alike,  we  should 
not  use  F  x  as  a  notation  for  each  of  them. 

133.  We  have  also  given  in  the  first  Section  (Art.  11) 
several  equations  which  are  generated  from  supposed  con 
ditions  in  a  problem,  in  such  a  way  that  no  function  of  x 
stands  equated  with  a  single  equivalent  in  y.  Such  are 
implicit  or  implied  (functions  of  x)  =  y.  We  are  able, 
however,  in  simple  cases  of  this  kind  to  elaborate  from  the 
equation  the  explicit  (function  of  x)  =  y.  When  we  may 
not  readily  do  this  or  cannot,  we  allude  to  the  condition 


IMPLICIT   FUNCTIONS.  109 

after  transferring  both  members  to  one  side  of  the  equation 
(leaving,  of  course,  0  in  the  other)  as  a  (function  of  x  and 
y)  =  0,  or  as  F  (x,  y)  —  0,  and  here  it  is  understood  that 
any  and  all  terms  consisting  of  isolated  constants,  are 
included  in  the  designation  ;  it  is  enough  that  x  and  y  are 
somehow  present.  Of  course  in  just  this  notation,  viz., 
F  (x,  y),  we  find  no  algebraic  association  of  quantities ;  the 
whole  expression  is  the  equivalent  of  a  sentence  in  lan 
guage  ;  the  comma  between  x  and  y  is  intended  to  assist  in 
dispelling  any  notion  of  a  product  of  x  by  y,  or  of  any 
specific  algebraic  association. 

134.  The  general  equation  of  the  Second  Degree  (Art. 
43)  shows  the  most  involved  relation  possible  for  this  De 
gree  between  x  and  y ;  A,  B,  (7,  etc.,  being  supposed  to 
be  constants,  this  equation  may  be  cited  as  F  (x,  y)  =  0. 

135.  There  is  a  peculiar  significance  in  F  (x,  y)  being 
equated  with  0  ;  it  could  not  be  equated  in  general  nota 
tion  with  any  thing  else,  for  every  case  of  such  use.     Now 
in  F  (x,  y)  =.  0  there  is  still  the  independent  variable  «, 
and  the  dependent  variable  y,  as  ever ;  they  are  mutual 
dependents,  intended  as  ever  preserving  that  combination 
of  themselves  with  each  other,  and  perluips  with  constants 
equal  to  zero. 

136.  But  if  x  and  y  each  stand  for  quantities  which  are 
lo  vary  independently,  the  value  0,  which  in  any  such  par 
ticular  case  cannot  vary,  should  be  replaced,  say  with  2,  to 
denote  the  corresponding  values,  so  that  we  shall  have 

F  (x,  y)  =  z. 

137.  An  extension  of  this  notation  to  an  implicit  function 
of  two  independent  variables,  gives  us 

F  (x,  y,  z)  =  0. 
10 


110  DIFFERENTIAL   CALCULUS. 

138.  If  there  be  three  independent  variables,  the  nota 
tion  would  be 

F  (x,  y,  z)  =  10, 

and  so  on.     This  is  the  system  of  symbols  for  the  general 
designation  of  implicit  functions  of  variables. 

139.  In  algebraic  equations  with  two  or  more  unknown 
quantities,  in  which  conditions  are  furnished  for  the  nota 
tion  of  the  calculus,  viz.  : 

Fx+fy+f  z  =  a, 


etc.,  etc.,        =  c, 

means  are  supposed  to  be  furnished  by  the  independence 
of  these  equations,  for  the  elimination  successively,  say  of 
all  functions  of  z,  then  of  y,  till  we  derive  some  (function 
of  x)  =  some  combination  of  a,  5,  c,  etc.,  thence  to  find 
x,  thence  y,  thence  z.  But  such  functions  must  be  of  the 
simplest  form  to  render  this  possible. 

140.  We  proceed  to  «how  that  it  is  not  necessary  to  de 
duce  in  form  an  explicit  function  of  an  independent  varia 
ble  from  the  implicit   state,  that  we   should  be  able  to 
differentiate   such   explicit   function    as  to  its  dependent 

variable  ;  that  is,  we  may  find  —  from  F  (a?,  y)  =  0  with- 

d  x 

out  finding  /"a;  =  y. 

141.  It   will   have  been  understood   in  differentiating 
a  case  of  F  x  =  y,  and  making  the  notation  d  F  x  =  d  y, 
the  terms  of  an  actual  particular  function  taking  the  place 
of  F  x,  that  we   have   given  two   names  for  the   same 
amount.     The   function    is   one.     We   pass   the    sign    of 
equality,  and  express  by  d  y  what  is  also  expressed  in  the 
other  member  of  the  equation.     This  consideration  would 


DIFFERENTIATION   OF  IMPLICIT   FUNCTIONS.          Ill 

at  once  authorize  us  to  transpose  y  previously,  so  that 
from 

Fx  —  y  =  0, 

we  should  have         d  F  x  —  d  y  =  0. 

Again,  there  is  obviously  no  reason  but  that  of  simplicity 
why  we  equate  F  x  with  y  rather  than  with  a  y  or  y  2,  i.  e., 
with  some  function  of  y.  If  u  be  such  a  function  of  y  in 
amount,  we  might  have  for  differentiation  of  x  with  respect 
to  y  as  a  dependent  variable,  such  a  case  as 

b  ic3  -f-  a;  =  y2  —  a  y  =.u\ 
differentiating  fy  =  u,  we  have 

^y  dy  —  a  dy  =  d  u, 

where  y  may  be  independent  with  respect  to  w,  but  by 
hypothesis  y  may  be  still  also  dependent  with  respect  to  x. 
Since  F  x  is  equated  with  w,  we  are  entitled  in  differen 
tiating  it  with  respect  to  w,  to  the  following  equation,  and 
in  doing  this  we  do  not  recognize  y  as  other  than  a  con 

stant  : 

d  Fx  =  3  b  x*  dx-\-  dx  =  du, 

so  that  we  may  infer  that  we  are  entitled  to  the  following 
equation  (remembering  that  the  differential  of  fy  depends 
ony): 

3bx*dx-\-dx  =  Zydy  —  a  dy  =  du: 
from  which  we  are  able  to  deduce  an  expression  for  —  , 
as  follows  : 


y  dx  dx        dx' 

dy  du  du    .    du 


112  DIFFERENTIAL   CALCULUS. 


Now,  if  instead  of  the  above  case  of  F  x  —  /?/, 
we  had  had  the  same  in  the  form  off  (x,  y)  =  0  =  u  , 

that  is, 

&  a?3  —  ys  +  a?  +  a  y  =  0  =  V, 

there  occurs  nothing  that  would  alter  the  result  for 
—  .  Here  we  have  given  an  accent  to  u  for  denoting  that 

'  d  x 

u  is  used  for  f  (aj,  y)  instead  of  for/y,  as  before.  But 
why  do  we  use  u'  at  all  since  we  have  zero  also,  and  since 
d  u'  is  manifestly  0  ?  The  answer  is,  u'  =  0  and  d  u'  •=.  0 
always;  but  we  are  proceeding  to  make  the  fraction—, 
and  instead  of  —  or  —  we  find  it  more  useful  to  use  some- 

dx         dy 

thing  that  will  preserve  a  special  relation  to/'  (x,  ?/),  and 
be  a  means  of  discrimination  whenever  we  should  have 
associated  in  an  investigation  implicit  functions  which  we 
might  be  obliged  to  discriminate  as/1  (#,  y)  =.  0  =  u,  and 
f(z,w)=Q  =  v,  etc. 

Zero  does  not,  in  the  ordinary  use  as  0,  preserve  its 
record  and  identity  ;  does  not  take  account  of  its  factors  ; 
if  /  (x,  y}  =  0,  then  10  X  /  (#,  y)  =  0,  but  in  the  use  of 
u  we  should  have  10  u. 

It  is  an  important  principle  in  differentiation  to  observe, 
that  we  allow  no  loss  of  quantities,  such  as  the  factors  of 
zero,  and  receive  none  such  of  which  we  do  not  make  a 
record. 

We  may  generally  place  (/  (#,  y)  =  0)  =  w,  for  the  pur 
pose  of  differentiation,  without  the  accent. 

Let  the  next  be  a  more  intimately  connected  case  of 
/  (*»  y)  =  0,  viz.  : 

a  x  y*  -\-  b  x*  y  -f  a2  +  y2  -\-  c  =  0  =  u, 
.:  2  a  x  y  dy  -\-  ay*  dx  -\-  b  x*  d  y  -f-  2  b  y  x  dx  + 


DIFFERENTIATION   OF   IMPLICIT   FUNCTIONS.          113 


d  u 

now,  Zaxy-\-bx2-\-2y  =  —  ; 

dy 

dy  du    ^    du 

d  x  ~          dx    '    d  y' 

In  finding  —  we  differentiate  u  with  respect  to  just  the 

variable  y. 

If  the  above  particular  demonstration  would  authorize  a 
general  rule,  such  rule  would  be  as  follows  :  but  a  general 
demonstration  would  be  quite  abstruse.  While,  then,  the 
rule  is  merely  stated,  it  is  recommended,  for  the  purposes 

d  'u 

of  the  following  Section,  to  work  out  the  results  for  — 

in  a  manner  similar  to  the  above,  by  differentiating  terms 
in  F  (x,  y)  =  0,  when  the  following  rule  will  be  found  to 
be  a  declaration  of  each  result  : 

143.  Whenever  we  have  a  case  that  can  be  cited  as 
f  (x,  y)  =  0,  and  we  wish  to  derive  —  ,  we  differentiate 

d  x 

the  expression  as  if  y  were  a  constant,  and  divide  the 
coefficient  so  obtained  by  the  coefficient  obtained  from  dif 
ferentiating  the  same  expression  on  the  supposition  that  x 
is  constant,  and  then  change  the  sign  of  the  fraction  so 

obtained  ;  this  fraction  is  —  . 

d  x 

PROBLEMS. 

1.  In  the  implicit  function  of  the  variable  x  in  &2  + 
10  y  -)-  5  x  =  0,  required  —  .  Ans.  —  =  --  . 

10* 


114  DIFFERENTIAL   CALCULUS. 

2.  Required  --  in  x*  +  y*  —  50  =  0. 

dy  X 

Ans.  ~  =       -  =  ± 

dx  y  V50 

This  case  shows  an  obvious  reduction  of  an  implicit  dif. 
coef.  reduced  to  an  explicit  one  ;  the  previous  case  did  not 
require  such  reduction.  In  general,  the  explicit  dif.  coef. 
may  not  be  easily  found,  or  may  not  be  needed ;  all  the 
purposes  of  a  dif.  coef.  being  subserved  by  the  implicit 
dif.  coef. 

3.  Required  —  in  a2  +  y2  —  m  x  y  —  81  =  0. 

dy         my  —  Zx 

Ans.  —  = . 

d x        1y — m x 

4.  Required  —  inse3  —  3  c  #  y  -|-  y  3  =  0. 

dy         cy 

Ans.  —  = 


dx        y*—cx 


5.  Required  —  in  (x  —  a)  2  +  (y  —  b)  2  =  0. 

(I  X 


dy         x—a 

Ans.  --  =  -  - 

dx        y  —  b 


6.  Required  d-  in  24  a2  y  —  y*  +  10  x  =  0. 


d  y  48  x  y  +  10 


144.  It  is  scarcely  necessary  to  say  the  successive  differen 
tiation  of  /  (a?,  y)  =  0,  relatively  to  y  as  dependent  and  x 
as  independent  variable,  may  be  performed  ;  and  that  Tay 
lor's  Theorem  and  the  theory  of  maxima  and  minima  are 
available  for  F  x  =  y  in  the  cases  given  as/  (a;,  y)  =  0. 

145.  But  f  (x,  y)  =  0  is  constant,  and   can   have   no 


CONTRADICTORY   EQUATIONS.  115 

maximum  or  minimum,  as  is  very  obvious,  and  in  mak 
ing  this  remark  we  are  not  alluding  to  f  x  involved  in 
/(oj,y)=0. 

140.  In  a  treatise  of  algebra  the  following  are  given  as 
illustrations  of  two  equations  that  are  contradictory,  be 
cause  the  same  value  for  x  is  not  deducible  from  each : 

1.  3  x  =  60 ; 

2.  £  x  =  20. 

But  suppose  we  proceed  to  consider  them  in  this  way : 

3.  Zx  —  60  =  0'; 

4.  £ce  —  20  =  0"; 

5.  .-.  3  x  —  60  =  %  x  —  20  •'•  a  =  16  ; 

whence  the  question  arises,  what  is  there  illogical  about 
such  a  course  of  proceeding?  although  the  result  is  not 
compatible  with  x  =  20  or  x  =  40,  as  would  be  severally 
deduced  from  (1.)  arid  (2.).  The  answer  is,  that  x  in  (3.)  is 
an  alien  from  x  in  (4.),  and  by  independence  it  renders 
F  x  =  0  in  (3.),  and /"#  =  ()"  in  (4.)  ;  or  35,  for  the  instant 
considered  a  variable,  produces  the  value  0'  in  (3.),  and  0" 
in  (4.),  by  an  independent  law  of  change.  In  regard  to 
equation  (5.),  we  can  say  that  the  value  x  =.  16,  is  such  as 

will  truly  render  Fx=fx.     Now, =  ,  the 

d  x  d  x 

functions  do  not  change  at  like  rates.  Indeed,  the  zero  0' 
in  (3.)  and  0"  in  (4.)  are  not  produced  from  like  elements, 
and  are  not  compatible  with  each  other,  and  not  equal. 
No  better  proof  could  be  desired  of  the  statements  made 
in  previous  Sections  in  regard  to  the  diverse  values  of 
zero,  as  dependent  on  distinct  origins. 


116  DIFFERENTIAL   CALCULUS. 

Suppose  we  have  the  two  algebraic  equations, 

1.  3as  =  0',   .-.  a:  =  0'"; 

2.  £a;  =  0",/.a;  =  0''"; 
whence  if  0'  =  0",  we  have 

3.  3x  =  £x; 
dividing  by  cc,  3  =  ^ ; 

or,  3  «;  _  ice  —  0'  —  0"; 

0'  — 0' 


/.  x  = 


3  -a 


These  results  are  only  to  be  reconciled  by  the  diverse 
values  of  zero,  as  well  as  of  x  when  at  zero. 

Suppose  next  we  have  the  two  equations, 

1.  3  x  —  60  =  0   /.a  =  20; 

2.  (  3  aj  =  0'. •.»  =  <)"; 
whence  if  0  =  0',  we  have 

3.  3  x  —  60  =  3  x ; 

60 

,.*  =  -  =  <*. 

Now,  in  no  case  are  the  equations  (3.)  absurd  in  nature, 
but  the  mode  of  making  them  is  not  consistent  with  (1.)  and 
(2.)  in  the  three  supposed  cases. 

147.  In  a  former  part  of  this  section,  the  nature  and  the 
differentiation  of  a  (function  of  x  and  y)  =  0  were  dis 
cussed  :  there  might  have  been  suggested,  in  that  connec 
tion,  the  question,  if  we  have  two  functions  of  #,  F  x  and 
which  might  be  equal,  whether,  when  they  are,  their 


DIFFERENTIALS   OF   EQUATIONS.  117 

differentials  are  also  necessarily  equal?  But  they  are  not. 
This  being  contrary  to  what  a  superficial  view  would  lead 
us  to  adopt,  is  worthy  of  a  statement  of  the  reasons. 

In  the  simple  condition  given  of  F  x  =  f  a-,  in  the 
maintenance  of  which  x  in  F  x  has  a  particular  value,  it  is 
pure  presumption  to  suppose  that  this  is  the  same  as  that 
of  x  in/a-,  at  which  x  in  fx  gives  fx  the  same  value. 

In  the  case  of  F  x  =  y,  we  call  x  an  independent  varia 
ble  ;  also  in  the  case  of  f  x  —  y ',  we  call  this  x  an  inde 
pendent  variable.  Now  in  the  accommodation  of  the 
particular  condition  of  y  =  y',  x  in  Fx  and  x  iufx  must 
retain  their  independence  as  ever,  as  more  likely  to  accom 
modate  the  condition,  which  a  simple  example  should  show. 
Let  F  x  be  x2  —  40  x  +  1802,  and/a  be  80  x  —  x*,  then 
if  F  x  =fx,  we  have 

a  2  _  40  x  _|_  1802  =  80  x  —  x* ; 

but  we  cannot  deduce  any  common  value  of  x  in  F  x  and 
f  x,  by  which  this  possible  equation  is  sustained. 

When  we  come  to  the  dif.  coefs.  ofFx  and  fx,  they 
cannot  be  inferred  to  be  necessarily  equal  for  the  mere 
reason  that  the  functions  from  which  they  have  been  de 
rived  happened  to  be  such  that  they  might  have  a  com 
mon  value,  but  nothing  else  of  nature  in  common.  Dif 
ferential  coefficients  show  the  nature  of  functions  through 
all  values.  * 

The  exhibition  of  many  cases  of  functions  of  x  and  their 
being  equated  with  y,  tends  to  create  an  illusion  as  to  their 
entire  incompatibility  when  they  enter  into  random  asso 
ciation  as  above.  They  should  at  once  be  expressed  in 
proper  language  for  such  association,  as  F  x  =  y,  f  z  =  v, 

'w  =  u,  etc. 


118  DIFFERENTIAL   CALCULUS. 


SECTION  XVI. 

PROBLEMS  WHICH  MAY  FURNISH  IMPLICIT  FUNCTIONS 
OF  ONE  VARIABLE,  AND  CASES  OF  THEIR  MAXIMA 
AND  MINIMA. 

1.  A  drover  bartered  160  head  of  cattle  for  sheep :  the 
number  of  sheep  obtained  in  the  exchange  was  found  to 
be  40  times  the  dollars  allowed  for  the  value  of  a  sheep. 
It  is  required  to  determine  the  law  or  rule  by  which  the 
value  of  one  of  the  cattle,  in  number  of  dollars,  will  vary 
in  the  fulfilment  of  these  conditions,  compared  with  the 
varying  value  of  a  sheep. 

Let  y  —  number  dollars  for  1  of  the  cattle, 

and  x  •=.  number  dollars  for  1  of  the  sheep ; 

then      40  x*  =  160  y,  or  40  a2  —  160  y  =  0  ; 
mxdx—  160  dy\ 

,.  d-y  =  j  * 

dx         <* 

Ans.  The  number  of  dollars  value  of  one  of  the  cattle, 
tends  to  increase  as  many  times,  or  as  much  of  a 
time,  the  number  of  dollars  value  of  a  sheep,  at 
any  supposition  for  either,  as  £  of  what  the  number 
of  dollars  value  of  a  sheep,  at  any  such  supposition, 
may  be. 

2.  A  fruit-seller  carried  a  certain  number  of  bushels  of 
fruit  to  market,  which  he  sold,  and  expended  $8.50  of  the 
proceeds  for  grain,  when  he  found  that  he  had  in  money, 
unspent,  the  value  of  5  bushels  of  the  fruit.     If  there  had 
been  22  bushels  of  the  fruit,  by  supposition,  and  then  just 


MAXIMA,  ETC.,  OP   IMPLICIT   FUNCTIONS.  119 

an  increase  in  that  number  be  suggested,  what  effect  will 
this  suggestion  have  on  their  change  in  value  per  bushel, 
in  still  fulfilling  the  conditions  of  the  problem  ? 

Let  x  =  the  number  of  bushels  of  fruit, 

and  y  =  the  number  of  cents  per  bushel ; 

/.  x  y  —  850  =  5  y ; 
:.  ydx-\-xdy  =  5dy, 

dy  _  850  850 

"Tx~      ~  (*  — 5)2  ~      ~  289' 

Hence  the  price  in  cents  must  commence  to  diminish  at 
the  rate  of  2f  ff  times  the  number  of  bushels  should  be 
supposed  to  increase.  This  incipient  ratio  is  expressed  in 
these  units,  but  would  itself  vary  during  the  change  of  as 
much  as  the  whole  unit,  one  bushel. 

3.  (a.)  A  courier  rode  30  hours  in  all,  but  successively 
on  a  gray  and  a  red  horse ;  the  number  of  miles  on  the 
gray  one  was  7  times  the  miles  he  went  per  hour  on  the 
red  one ;  the  number  of  miles  on  the  red  one  was  9  times 
the  miles  he  went  per  hour  on  the  gray  one :  it  is  required 
to  determine  how  the  miles  per  hour  on  the  gray  one  must 
be  inferred  to  change  relatively  to  the  number  per  hour  on 
the  red  one,  on  any  suggestion  of  change  of  number  of 
miles  per  hour  on  the  red  one,  when  either  is  at  any  possi 
ble  rate  allowed  by  the  conditions. 

Let  x  =  miles  per  hour  on  red  horse, 

and  y  =  miles  per  hour  on  gray  horse  ; 

d  x 

:.  —  =  number  hours  on  red  horse ; 
9 

by 

and  —  =  number  hours  on  gray  horse ; 


X 


120  DIFFERENTIAL   CALCULUS. 


*-  4.779)  =  - 


Whence  it  appears  that  y,  at  whatever  value  it  may 
have,  will  increase  minus  £th,  or  minus  3.08  times  as 
fast  as  x.  Since  —  is  constant,  and  cannot  —  0,  there  is 
no  limit  to  y.  Because  —  is  constant,  there  is  no  limit 

dy 

to  greatness  of  x.     Since  in 

7  a;2  +  9  y2  —  30  x  y  =  0, 

if  y  =  0,  x  =  0,  there  is  not  necessarily  any  distance 
attained  by  the  courier,  whether  he  sits  on  the  horse's  back 
30  hours,  or  any  other  number  of  hours,  and  the  word  ride 
seems  to  fail  in  practicability  in  that  condition.  But  his 
rate  of  motion  may  be  indeterminately  great. 

(b.)  A  courier  rode  30  hours  in  all,  but  successively 
on  a  gray  and  a  red  horse  ;  the  number  of  miles  on  the 
gray  one  was  7  times  the  miles  he  went  per  hour  on  the 
red  one,  and  1  mile  more  ;  the  number  of  miles  per  hour  on 
the  red  one  was  9  times  the  miles  per  hour  on  the  gray  one  : 
it  is  required  to  determine  how  the  miles  per  hour  on  the 
gray  one  must  be  inferred  to  change  relatively  to  the  num 
ber  per  hour  on  the  red  one,  on  any  suggestion  of  change 
of  number  of  miles  per  hour  on  the  red  one,  when  either  is 
at  any  possible  rate  allowed  by  the  conditions,  and  whether 
there  are  maxima  of  miles  per  hour  on  each  horse. 


MAXIMA,  ETC.,   OF  IMPLICIT   FUNCTIONS.  121 

It  will  be  ascertained  that  the  words  in  this  problem, 
"  and  1  mile  more"  change,  very  materially,  all  the  charac 
teristics  of  the  results  of  the  solution,  in  comparison  with 
problem  (a.). 

4.  (a.)  A  caterer  having  9  dollars  in  gold  and  a  num 
ber  of  dollars  in  silver,  purchased  at  a  market  90  quails 
and  as  many  more  quails  as  he  had  dollars  in  silver,  of  such 
value  each  quail,  that  his  silver  alone  would  have  paid  for 
144,  and  then  invested  the  balance  of  his  money  in  as 
many  pigeons  as  quails   already  bought.     Required   the 
number  of  dollars  in  silver  if  his  quails  cost  the  greatest 
possible  sum  each ;  and  required  that  sum. 

Ans.  18  dollars,  and  12 £  cents  each  pigeon. 
(b.)  Required  what  two  several  numbers  of  dollars  he 
may  have  had  in  silver  that  the  quails  alone  should  have 
exhausted  just  all  his  money. 

The  conditions  of  the  above  problem  (a.)  may  be  com 
pared  with  those  of  Problem  27,  in  Section  XII. 

5.  A  man  purchased  a  commodity  known  as  ^4,  for  which 
he  paid  as  many  cents  a  pound  as  it  weighed  pounds,  and 
the  number  of  pounds  of  A  differed  by  10  from  the  num 
ber  of  pounds  of  the  commodity  known  as  J?,  which  was 
his   second   purchase.     The    next    day  he  purchased  the 
commodity  distinguished  as  C,  at  the  same  number  of  cents 
a  pound  as  there  were  pounds,  and  its  number  of  pounds 
differed  by  15  from  that  of  the  number  of  pounds  of  the 
commodity  known  as  7>,  his  fourth   and  last  commodity 
purchased.     But  we  know  of  no  guarantee  that  each  of 
the  commodities  A  and  C  had  weight,  though  it  may  be 
inferred  that  one  of  them  must  have  had.     He  paid  for  A 
and   C  81  cents.     It  is  required  to  determine  the  possi 
ble   range    of  relative    weights    of  £    and   _Z>,   and  the 

11 


122  DIFFERENTIAL   CALCULUS. 

greatest   and   least   weights   of  each  of  these  two  com 
modities. 

Ans.  Limits  of  .7?,  1  and  19  pounds ;  of  J9,  6  and  24 
pounds. 

6.  (a.)  A  furrier   purchased  two   lots    of  furs :    when 
asked  what  he  paid  per  pound  for  the  kind  A,  he  said,  this 
purse  in  my  hand  contains  99  dollars  in  coins ;  if  I  take 
out  as  much  value  as  I  paid  per  pound  for  the  kind  J3^  and 
repeat  this  as  many  times  as  I  take  out  dollars  each  time, 
and  if  I  then  take  the  money  left  in  the  purse  and  divide 
it  into  as  many  piles  as  I  put  dollars  in  a  pile,  one  of  these 
piles  is  the  price  I  paid.     When  asked  how  many  pounds 
he  purchased  of  A,  he  said  2$-  pounds  more  than  I  just 
took  out  dollars  out  of  the  purse  at  once,  and  the  amount 
of  money  I  expended  for  each  lot  of  fur  was  alike.     Re 
quired  a  limit  of  the  price  of  I>. 

(b.)  If  we  adopt  the  principle  of  assuming  a  quantity 
for  .Z?,  with  the  intention  afterwards  of  exchanging  it  for  a 
quantity  the  least  greater,  at  what  amount  of  B  are  we 
stopping  in  the  consideration,  when  the  suggested  increase 
involves  the  most  rapid  decrease  of  its  price  per  pound. 

7.  (a.)  A    counterfeiter,  escaping  arrest,  rode   a  black 
horse  as  many  hours  as  miles  per  hour,  when  he  exchanged 
and  rode  a  white  horse  as  many  hours  as  miles  per  hour. 
At  the  instant  he  commenced  riding  the  white  horse,  an 
officer  commenced  a  pursuit  from  the  original  point,  at  six 
fifths  of  the  speed  of  the  black  horse.     When  the  fugitive 
is  done  with  the  white  horse,  he  is  just  81  miles  from  the 
officer,  at  which  point  he  is  arrested  through  the  aid  of  the 
telegraph.     The  owners  of  the  two  horses  which  had  been 
pressed  into  the  above  service  by  the  fugitive,  on  recover 
ing  them,  were  anxious  to  know  which  animal  had  been 
hardest  and  longest  driven.    It  is  required  to  determine  that 
particular  condition,  according  to  which  either  horse's  ser 
vice  was  the  greatest  the  conditions  admit  of. 


MAXIMA,   ETC.,   OP   IMPLICIT   FUNCTIONS.  123 

Let  x  =  black  horse's  hours  and  miles  per  hour, 

and  y  •=.  white  horse's  hours  and  miles  per  hour, 

and  c  =  six  tenths ; 

then    2  c  x  y  =  the  officer's  travel  in  miles  ; 
then  x2  +  y2 —  Zcxy  —  81  =  0;  (1.). 

dy         cy  —  x 

.'.  —  = =  0  when  11  •==.  max. ; 

dx         y  —ex 

.'.  c  y  —  x  =  0  and  y  =  —  x ; 

3 

substituting  this  value  of  y,  for  y  in  (1.), 
x  =  +    6for--    6f, 

Discarding  the  negative  values  of  x  as  impracticable,  for 
an  event  cannot  take  place  through  negative  time,  we 
wish  to  know  whether  the  positive  value  of  x  gives  a 
maximum  or  minimum  for  y,  by  examining  the  — - ; 

^  2  V         (y  —  c  x)  (c  d  y  —  d  x)  —  (c  y  —  x)  (d  y  —  c  dx) 
d  x  (y  —  c  x) 2 


.         dy 
since  —  =  0  ; 

dx 


d2  y 

:.  --  <^  0  /.  y  when  ll£  —  maximum. 
It  is  to  be  remembered  that  —  ^  in  the  general  state  is 


124  DIFFERENTIAL   CALCULUS. 

the  full  one  foregoing ;  for  —  has  been  made  special  when 

d  x 

made  =  0. 

(b.)  How  many  miles  per  hour  do  the  conditions  require 
the  white  horse  to  travel  when  the  black  one  travels  9  ? 

Ans.  lOf  or  none. 

(c.)  How  many  miles  per  hour  do  the  conditions  require 
the  white  horse  to  travel  when  the  black  one  travels  12  ? 

Ans.  The  supposition  is  forbidden  ;  so  great  a  value  for 
a;  as  12  is  imaginary. 

(d.)  What  two  values  satisfy  the  conditions  for  the 
white  horse's  speed  if  the  black  one's  is  made  10  miles  an 
hour.  Ans.  10?\  and  l^J  very  nearly. 

It  may  be  interesting  to  notice  that  the  maximum  speed 
and  duration  of  service  of  either  horse  is  not  that  which 
most  favored  the  other,  because  the  distance  executed  by 
the  fugitive  is  not  an  absolutely  determinate  amount.  The 
greatest  distance  attained  by  the  officer  is  evidently  when 
x  •=.  y  in  2  c  x  y,  which  occurs  when  x  =  y  =.  10.035  miles 
or  hours ;  and  since  the  fugitive  went  81  miles  more,  the 
value  of  x  and  y  =  10.035  gives  the  greatest  associated 
service  of  each  horse. 

8.  Tradition  says  that  a  certain  king  erected  a  solid 
structure  of  stone  20  (a)  rods  long,  its  width  and  height 
alike ;  that  his  successor,  or  the  second  king,  erected  two 
structures,  the  one  12  (b)  rods  long,  the  height  and  width 
alike,  the  other  12  (b)  rods  long,  8  (c)  rods  wide,  and  as 
high  as  the  one  last  mentioned.  His  successor,  the  third 
king,  converted  all  of  the  three  previous  structures  into 
two  of  his  own,  the  one  a  complete  cube,  the  other  8  (c) 
rods  wide,  and  of  that  uniform  height  of  all  the  structures 
of  the  second  and  third  kings,  and  as  long  as  high,  it  being 
understood  that  the  mere  names,  width  and  length,  are 
interchangeable  when  necessary. 


MAXIMA,    ETC.,    OF    IMPLICIT   FUNCTIONS.  125 

The  question  is  the  height  of  the  first  king's  structure, 
what  it  might  have  been,  and  whether  it  existed  at  all,  if 
the  second  king  built  his  less  than  12  rods  high. 

The  negative  values  of  any  quantity  being  impracticable 
within  this  problem,  the  study  of  the  function  at  negative 
values  of  the  variables  bring  out  very  peculiar  properties, 
which  are  best  shown  by  a  diagram. 

9.  (a.)  For  a  certain  purpose  two  cubical  cisterns,  A 
and  _Z>,  are  needed,  and  one  a  rectangular  prism,  C.     The 
cubical  cisterns,  A  and  B,  are  to  have,  combined,  the  same 
capacity  as  C.     (7  must  have  one  linear  dimension  15  feet, 
another  the  same  as  that  of  cistern  A^  the  other  the  same 
as  the  cistern  JB.     A  well-informed  contractor  agrees  to 
make  the  cistern  JS  for  a  stipulated  sum.     The  other  party, 
designing  to  receive  the  largest  possible  cistern  for  the  fixed 
sum,  studies  to  vary  the  size  of  the  cistern  A.     Required 
to  know  all  the  dimensions  of  the  three  cisterns  when  the 
one  here  contracted  for  is  the  largest  possible. 

Ans.  A,  5  X  2  *  =  6.299  feet. 
B,  5  X  4*  =  8.025  feet. 

(£.)  Since  either  of  the  cubic  cisterns  can  have  a  linear 
dimension  8.025  feet,  if  one  be  made  only  7  feet,  required 
the  linear  dimensions  of  the  others. 

(c.)  If  the  two  cubic  ones  are  equal,  how  large  are 
they? 

(d.)  Do  all  the  cisterns  become  of  no  capacity  if  either 
one  does  ? 

10.  (a.)  A  balloonist  being  asked  to  give  some  account 
of  the  heights  and  distances  of  his  latest  ascension,  said 
that  y,  conditioned  as  follows,  was  his  height  in  miles  every 
where  on  his  voyage  over  the  level  country,  while  x  was 
his  distance  by  horizontal  measure  in  miles  from  his  point 
of  beginning  to  rise,  viz. : 

48  y  =  192  x  —  88  a2  +  16  x3  —  x4  ; 
11* 


126  DIFFERENTIAL   CALCULUS. 

in  which  we  have  F  y  =fx,  and  implicitly  y  =f  x,  and 
in  this  simple  nature  of  F  y  it  is  evident  we  may  easily 
express  y  — /'  ce,  which  is  the  function  of  x  that  is  signifi 
cant  in  the  problem.  Required  the  prominent  points  in 
the  history  of  heights  and  distances  of  the  voyage,  such  as : 

(b.)  What  was  the  distance  made  at  the  landing  ? 

(c.)  What  and  where  was  his  greatest  height,  if  there 
was  any  greater  than  all  others  ? 

(d.)  To  determine  if,  after  first  descending  any,  he  after 
wards  ascended,  and  where,  and  being  how  high  he  com 
menced  any  second  ascent  ? 

(e.)  To  determine  if,  in  rising  the  second  time,  he  went 
up  to  the  same  greatest  height  to  which  he  had  ascended  ? 

(/!)  How  far  apart  were  any  two  places  at  which  he 
began  to  descend,  measured  on  the  ground  line  ? 

(g.)  Where  in  the  voyage  he  rose  most  directly  up 
ward  ? 

(A.)  How  far  from  the  point  of  starting,  ground  measure, 
was  he  at  each  of  his  greater  heights  ? 

(i.)  If  he  had  gone  in  a  straight  line  in  the  direction  he 
first  started,  how  high  would  he  have  been  when  200  feet 
from  his  starting  point,  ground  measure  ? 

(j.)  Had  he  risen  out  of  the  earth  and  descended  into 
it  after  alighting,  would  the  function  indicate  the  law  of 
his  course  beneath  the  earth's  surface  ? 

(&.)  Specify  the  term  in  the  function  which  intimates  that 
he  must  rise.  Ans.  192  .:•. 

(7.)  Specify  the  term  which  intimates  that  he  must 
finally  come  down.  Ans.  —  a-4. 

(m.)  Specify  the  term  which  intimates  the  probable 
originating  of  a  second  place  of  rising ;  probable,  because, 
in  the  generalization  of  the  constants,  such  term  might  be 
neutralized  by  another  one  just  equal  to  it  in  value,  with 
an  opposite  sign. 

(n.)  To  what  distance  below  the  earth's  surface  does  the 


MAXIMA,   ETC.,   OF  IMPLICIT  FUNCTIONS.  127 

function  indicate  the  law  of  his  course,  including  the  con 
sideration  of  x  when  negative  ? 

(o.)  Is  there  any  thing  symmetrical  between  the  first  and 
last  halves  of  the  voyage  ? 

(p.)  If  the  sign  of  every  term  in  the  function  be  changed, 
is  the  voyage  indicated  as  a  dive  below  the  earth's  surface, 
and  a  final  emergence  from  it  ? 

(q.)  Then  what  are  the  presumptions  about  the  course 
before  and  after  diving  ? 

The  first  dif.  coef.  is  of  the  third  degree,  but  may  be  re 
solved  by  trials  with  integral  numbers  for  x. 

The  presumption  has  been  that  this  aerial  voyage  was 
performed  in  one  perpendicular  plane  ;  but  it  will  be  per 
ceived  not  to  be  essential  if  the  ground  line,  however  tor 
tuous,  is  considered  to  be  beneath  the  voyage  track,  and 
measurable  like  a  straight  line. 

(r.)  It  is  required  to  alter  the  ascertained  y  =ff  x  (per 
haps  by  factor  common  to  every  term),  so  that  the  greatest 
heights  may  be  expressed  as  at  3^-  miles,  and  all  other 
heights  in  proportion. 

(s.)  Need  such  a  change  alter  the  distance  of  the 
landing  ? 

(t.)  It  is  required  to  modify  y  =ffx  so  that  the  place 
of  the  landing  may  be  15  miles,  without  affecting  the 
heights  as  first  conditioned. 

(u.)  It  is  required  so  to  give  out  the  F  y  that  the  place 
of  landing  may  be  14  miles,  and  greatest  heights  2  miles 
in  the  same  connection. 

11.  (a.)  There  is  to  be  determined  the  size  of  a  square 
piece  of  land,  which  it  is  proposed  to  enclose  with  a  fence, 
at  the  cost  of  three  or  a  times  as  many  dollars  per  rod  in 
length,  as  is  the  worth  of  as  many  square  rods  of  the  land 
as  each  of  said  square  rods  is  worth  dollars ;  and  the  cost 


128  DIFFERENTIAL   CALCULUS. 

of  the  whole  fence  will  be  60  or  b  dollars  less  than  the 
the  worth  of  the  land,  and  the  land  the  smallest  possible. 

Let  y  =  the  number  of  square  rods, 

and  x  =  the  number  of  dollars  per  square  rod ; 

then  42/5  =  the  number  of  linear  rods  round ; 

and      4  a  y\  a;2  =  the  dollars  cost  whole  fence ; 
then  4  a  2/5  a2  —  xy  +  b  —  Qy  (1.). 

dy  8ay*x  —  y 

and  —  — 1  • 

dx         x—2ax*y    * 

the  numerator  being  =  0  in  case  y  =  max.  or  min., 

...  y  =  64  a2  x\ 
by  substitution  in  (1.) 


and  2/:=64«2    -  -     =64    -      =202.44; 

\32aV  \32/ 

/.  one  side  =  13.974  rods. 

But  to  ascertain  beyond  doubt  whether  we  have  cer 
tainly  either  a  maximum  or  minimum,  and  which,  we  must 
deter 
had: 


determine  whether  —  is  zero,  or  positive  or  negative  ;  we 


d  y  &  a  y*  x  —  y 


x         x  — 


dzy  _  (x  —  2ax2y~)X(8ayrfar  +  4aary~rfy  —  d  y) 
"  dx    '  * 


TWO   INDEPENDENT  VARIABLES.  129 


(x  — 

multiplying  terms  and  dividing  by  d  tc,  we  have, 

d*y         16  a2  z2  -f  y—  ±ay\x  dy 

' 


48  a2  b  I 
the  numerator  01  which  =  --  - 


and  therefore  second  dif.  coef.  is  positive. 
/.  y  is  found  at  a  minimum. 

.*.  x  and  y  are  determinate  by  the  concurrent  equations  ; 
4  a  ys  a;2  —  a;  y  -|-  ft  =  0, 
8  a  2/2  a;  —  y=zO. 

NOTE.  In  consequence  of  the  length  of  the  above  expression  for  ^J/,  we 
have  expressed  the  same  in  two  terms  having  a  common  denominator  ;  this 
accounts  for  the  signs  of  —  8  ay5  x  -j-  y- 

(b.)  Required  the  size  of  the  lot  when  the  least  number 
of  dollars  is  paid  per  square  rod  for  it,  and  what  that  num 
ber  of  dollars  would  be. 


SECTION  XVII. 

FUNCTIONS  OF  TWO  INDEPENDENT  VARIABLES  :  THEIR 
DIFFERENTIATION  AND  THEIR  MAXIMA  AND  MINIMA. 

148.  The  mode  of  notation  by  which  we  may  cite  a 
function  of  two  independent  variables  has  been  (Art.  136) 
shown  to  be  f  (x,  y)  =  z.  From  the  circumstance  of 


130  DIFFERENTIAL   CALCULUS. 

identity  we  must,  therefore,  have  df  (#,  y)  =  d  z.  Sinc.Q 
z  may  vary  on  a  variation  of  x ;  and  since  at  the  same 
time  y  need  not  vary  on  account  of  its  independence, 
we  express  this  variation  of  z  with  respect  to  x,  by  differ 
ential  coefficient,  as  —  ;  and  the  variation  of  z  with  respect 

dy  dz 

to  y  by  differential  coefficient,  as  —  . 

There  is,  then,  no  better  way  for  expressing,  beyond 
doubt,  the  whole  differential  of  z  with  respect  to  x  and  to 
y,  by  general  notation,  than  by  (d  z),  or 

dz  dz 

—  a  x  -\ d  v ; 

dx          r  dy     y  ' 

which  becomes  very  intelligible  if  we  remember  that  in  a 
common  case  of  f  x  =  y,  df  x  might  have  been  cited  as 

-  d  x,  but  which  was  unnecessary.     In  all  particular  cases, 
d  x 

however,  in  this  section,  we  shall  have  use  for  the  expres 
sion  of  only  the  dif.  coefs.  —  and  — .  Although  we  ex- 

dx      ,dy          dz     dz 

press  the  whole  differential  coefficient  of  z  as 1 ,  we 

d  x         dy 

use  the  signs  in  the  general  sense ;  particular  conditions 
may  render  either,  or  both,  negative  in  value,  although  the 
amount  of  the  change  of  value  of  x  and  of  y  may  be 
positive. 

149.  In  differentiating  a  (function  of  x,  y)  =  z,  we 
may  conveniently  express  the  dif.  coefs.  —  and  —  in  suc- 

d  x  d  y 

cessive  equations,  that  variable  being  considered  a  constant 
with  reference  to  which  we  are  not  differentiating  the  equa 
tion.  Their  algebraic  sum  is  the  total  differential  coeffi 
cient  required. 


TWO   INDEPENDENT   VARIABLES.  131 

1.  Required  the  whole  dif.  coef.  of  3  x^  y  -\-  x  =  z. 

Ans.  —  +  --  =  6  x  y  +  1  +  3  a;2. 

dx     •     dy 

2.  Required  the  whole  dif.  coef.  of  —  =  z. 

y 

d  z         d  z         y  —  x 
Ans. = . 

dx     '     dy  y'2 

3.  Required  (d  z),  or  whole  differential  of  -         -  =  z. 

A          d '  z  J        i    d z  J  /  7  \        y  dx  —  3y*  dy  —  xdy 

Ans.  — a  x  -4 ai/=(az)  =  —  -. 

dx  ' dy  (3y2  —  x)* 

4.  Required  the  whole  dif.  coef.  of  15701  xy-\-ax2  =  z. 

150,  It  is  useful  and  quite  important  to  extend  Taylor's 
Theorem  to  embracing  a  development  of  a  function  of  two 
independent  variables,  the  condition  being  that  each  varia 
ble  may  concurrently  take  an  increment  or  decrement,  or 
one  variable   an  increment  and  the  other  a  decrement; 
which  condition  must  embrace  the  case  of  variation  limited 
to  one  of  the  variables. 

151.  In  the  function  f  (x,  y)  =  z,  if  x  take  the  incre 
ment  A,  the  function  will  become/"  (x  -\-  A,  t/),  y  remain 
ing  unchanged,  since  it  is  independent  of  x :  then,  by  Tay 
lor's  Theorem, 

/(a;  +  A,2/)=S  +  ^A+g.;^+,etc.          (1.). 

But  if  y  also  take  an  increment  #,  then  z  in  the  above 
expression  becomes  changed  to 

.    d  z  d*  z        k*      ,     d*  z          £3 

z  H Jc  A . . k  etc.,     (2.). 

r  dy  dy*      l.2~dy*      1.2.3    ' 

so  that  in  place  of  z  in  (1.),  we  must  substitute  the  whole 
of  (2.),  and  in  doing  so  after  we  have  passed  2,  we  must 


182  DIFFERENTIAL   CALCULUS. 

for  —  put,   in    (1.),   the   dif.   coefs.   (with  respect  to  ic), 

d  x 

of  every  term  in  (2.)  ;  that  is,  we  must  substitute 


' 


.  -    , 

1.2    ' 


dx3    '       dx3  dx*         1.2 

and  so  on.  Before,  however,  making  these  substitutions 
for  convenience  only,  and  not  as  an  algebraic  act,  let  us 
agree  to  write 

d   d  z  d   d2  z 

"   for-^-*.   T^for-^, 


dydx  dx        dyzdx  dx 

P   d<lz 
and,  generally,  g      p  for 


153.   Hence  the  result  of  the  proposed  substitutions  in 
(1.),  will  give  us 

/(•,+  **?.+ *>•"•  + 

^2    7  ^22          A2        .      d*  Z  A3 

—  ^  J . . -[-,  etc. ; 

dx       ~  dx*     l.*~4x*     1.2.3 

dz  7  d*z  d*z         kh* 

—  Jc  -\ —   —  kh-\ -. K  etc. ; 

dy  dydx  ^dydx*      1.2    T 

d*  z       &2  d*z         k*h 

. . (-,  etc.; 

dy*     1  .  2    '    dy*dx     1.2 

d3  z 


[->  etc- 

dy3      1.2.3    ' 

+,  etc. 


TWO    INDEPENDENT   VARIABLES.  133 

The  foregoing  is  the  development,  required  by  Taylor's 
Theorem,  of  a  function  of  two  independent  variables,  on 
each  of  them  undergoing  a  change  of  value. 

153.  If  every  term  containing  k  as  a  factor,  disappear,  as 
when  k  should  be  zero,  then  the  development  reverts  to 
one  for  a  single  variable  h. 

151,  If,  however,  h  and  k  be  negative,  all  those  terms  in 
the  foregoing  development  where  h  or  k  occur  at  even 
powers,  will  evidently  be  positive,  the  others  where  h  or  k 
stands  without  the  other  will  be  negative  ;  but  since  the 
development  is  supposed  to  hold  also  for 

F  (x  —  h,  y  +  k), 
or  F  (x  +  h,  y  —  k), 

the  terms  in  which  h  and  k  are  factors  together  must  be 
ambiguous,  in  the  development,  for  it  must  be  doubtful 
whether  h  k  arises  from  —  A  X  —  k  or  from  -j-  h  X  -\-  k. 

155.  Whenever  z,  in  a  case  off  (#,  y)  =  z,  is  at  a  maxi 
mum,  we  must  have  the  condition 


and  consequently 
(±  if  A  ±  If  ft)  +  $  (llf  A.  ±  2  -^f-  A  ft  +  '-!-' 

\      dx  dy     J         2\dx*  dxdy  rdy* 

-f-  etc.  <  0. 

Now,  since  there  is  no  reason  why  h  and  k  in  the  above 
expression  may  not  be  alike,  or  each  A,  the  above  inequa 
tion  may  be  written 

/       dz       d  z\  ,     ,          /d2  z  d2  z         dz  z\ 

(±  r.±  #*+*  fir>  2  J^+,T>  +'etc"<0; 

12 


134  DIFFERENTIAL   CALCULUS. 

this  condition  being  similar  to  that  in  Art.  (102),  we  infer 
by  the  same  reasoning  that 

d  z       d  z 
±  —  ±  —  =  0, 

dx         dy 

which  cannot  be  for  both  the  signs  ±  unless 

d  z  d  z 

-  =  0  and  —  =  0  ; 

d  x  d  y 

it  being  observed  that  owing  to  the  independency  of  the 

d  z  d  z 

variables,  and^the  distinctive  terms  —  and  —  depending 
severally  on  those  variables,  we  are  not  entitled  to  election 
of  signs  like 


indeed,  one  of  these  terms  need  not  exist  (Art.  150,  153), 
but  the  demonstration  must  hold.  Whatever  term  contains 
d  y,  must  have  contained  k  as  factor. 

Expunging,  then,  from  the  last  inequation  the  terms  —  0, 
the  condition  of  z  a  maximum  is, 


and  in  the  case  of  z  a  minimum,  we  should  have  derived  in 
the  same  way, 


a 


Now  x  and  y,  and  therefore  z,  and  all  the  coefficients  in 
fulfilling  the  above  conditions,  have  determinate  values  ; 
it  is,  therefore,  determinate  which  of  the  last  two  inequa 
tions  prevails  in  any  given  case,  if  we  can  avoid  the  com- 


TWO    INDEPENDENT   VARIABLES.  135 

plexity  of  the  ambiguous  sign  ±.     Let  us  represent  the 
terms  within  the  parenthesis  by 

A  ±  2  B  +  <7, 


B2        B* 
adding  0  ==  —  —  —  to  the  terms  between  the  brackets,  the 

expression  becomes 


A 

where  the  binomial  is  certainly  plus,  so  that  (since  A  and 
C  agree  in  sign)  if 

f^  R  2 

->-, 

that  is,  if  A  C  —  B*  >  0, 

the  whole  expression  within  the  last  double  brackets  will 
agree  with  A  in  sign.     Hence,  if 


there  is  certainly  a  maximum  or  minimum,  the  former  if 
l~  I  <  0,  the  latter  if  —^  >  0. 

156.  A  function  of  two  independent  variables,  x  and  y, 
may  have  a  maximum  as  to  one  variable,  and  a  minimum 
as  to  the  other,  at  the  same  time  ;  or  a  maximum  or  mini 
mum  as  to  one  variable,  and  neither  as  to  the  other. 

157,  In  functions  of  two  independent  variables,  there  are 
evidently  eight  conditions  of  value  in  regard  to  possible 


136  DIFFERENTIAL   CALCULUS. 

related  positive  and  negative  signs  of  the  function  and 
each  respective  variable  (see  Art.  97),  that  is,  as  many  con 
ditions  as  the  associate  quantities  ±  2,  rb  cc,  db  y  can  be 
written  in  different  ways,  with  the  single  sign  plus  or  minus 
to  each  severally  and  independently.  When  either  2,  x,  or 
y  is  at  zero,  such  is  a  value  of  transition  from  one  sign  to 
the  other;  neither  sign  being  significant  when  placed 
before  zero. 


SECTION  XVIII. 

PROBLEMS  INVOLVING  FUNCTIONS  OF  TWO  INDE 
PENDENT  VARIABLES;  AND  CASES  OF  THEIR  MAXI 
MA  AND  MINIMA. 

1.  A  person  appropriated  one  day  10  dollars  in  payment 
for  provisions,  that  were  to  be  distributed  in  equal  portions 
to  some  needy  families  ;  the  next  day  he  benefited,  by  gifts 
of  clothing,  27  times  as  many  families  as  each  of  those 
families  of  the  previous  day  received  pounds  of  provis 
ions  ;  on  the  third  day  he  benefited,  by  fuel,  8  times  as 
many  families  as  those  provisions  had  cost  cents  per 
pound.  Required  the  smallest  number  of  families,  in  all, 
which  by  any  possibility  may  have  received  his  aid  on  the 
three  days. 

Let         x  =  number  of  pounds  to  a  family, 
and  y  =  value  in  cents  per  pound  of  the  provisions, 

and  z  =  the  whole  number  of  families  required ; 


MAXIMA,    ETC.,    WITH   TWO    VARIABLES.  137 

1000    ,  d  z 

••— ?;  +  2'  =  r,' 

.    __^_L8   =— . 
x  y2  dy 

d  z  d  z 

Now  in  case  z  =  max.  or  min.,  we  have  —  =  0  and  —  = 

dx  dy 

0,  from  which  we  derive  x  —  3.31,  y  =  11.17,  and  z  = 
215.71.  Now  to  determine  whether  this  is  a  maximum  or 
minimum  value  for  2,  we  have 

2000         d2  z 

x3  y         dx'2 

2000         d2  z 


1000  d2z 

and 


x2y2         dxdy' 
d2  z         d2y  d2z     . 

so  that  for         — -  X  ~a  —   -  —  >  0, 

d  x*         d  y  z         a  x  a  y 
2000         2000    ,      1000   > 

we  have  — -  X  — -  H — - — -  ^>  u  ; 

d2  z  2000 

also  >  0,  that  is,         >  0 ; 

d x2  x3  y 

hence  the  value  215.71  for  z  is  a  minimum. 

Rational  considerations  will  evidently,  in  the  above  case, 
enable  us  to  determine  whether  it  be.  a  maximum  or  a 
minimum  for  2,  if  there  be  but  one  of  them ;  for  in  the 
case  of 


we  see  at  once  there  is  no  limit  to  the  greatness  of  z  when 
either  x  or  y  becomes  excessively  great ;  and  -  -  in  posi- 
12*  Xy 


138  DIFFERENTIAL   CALCULUS. 

tive  values  of  x  and  y  is  never  negative,  however  great  x 
or  y  may  be,  and  consequently  however  small  the  term 
may  thus  be  rendered. 

2.  It  is  required  to  divide  the  number  48  into  three  such 
parts  that  the  continued  product  of  the  first,  the  second 
power  of  the  second,  and  third  power  of  the  third,  may  be 
the  greatest  possible.  Ans.  8,  16,  and  24. 

3.  A  contractor  agrees  to  fence  the  four  sides  of  a  rec 
tangular  field,  but  with  two  kinds  of  fence,  one  worth  78 
cents  per  rod,  the  other  worth  $1.25  per  rod,  and  opposite 
sides  to  have  a  like  fence ;  and  he  agrees  to  dig  out  rocks 
from  one  square  rod  of  the  field,  this  work  being  worth  at  the 
rate  of  $7912  for  the  whole  field.     Required  the  length 
and  width  of  the  field,  the  cost  of  digging  out  the  rocks, 
the  cost  of  the  whole  fence  on  the  sides  and  on  the  ends ; 
when  the  sum  of  money  that  pays  for  the  whole  is  the 
smallest,  and  required  that  sum. 

Ans.  In  part,  width  of  lot  12.585  rods. 

4.  A  manufacturer  of  tin  ware  agrees  to  construct  a  tin 
box  of  rectangular  sides  and  bottom,  and  without  a  top, 
and  to  hold  just  5^  cubic  feet,  with  the  least  sheet  tin 
possible.     Required  the  dimensions  and  surface. 

Let 'the  bottom  be  x  by  y,  then  the  height  —  —  ;    then 

xy 

if  z  =.  the  surface,  we  have 

11       11 

z  =  xy  +  -  +  -- 

Ans.  x  must  =  y  =  (11)*  =  2.224  feet,  and  height  = 
£  (11)1  =  1.112  feet,  /.  the  box  is  one  half  of  a  cube 
cut  parallel  to  the  bottom.  But  the  whole  cube 
might  be  cut  any  how  by  a  plane  through  its  centre, 
without  a  variation  of  the  amount  of  surface  or  of 
the  contents. 


MAXIMA,    ETC.,   WITH   TWO    VARIABLES.  139 

5.  A  miner  in   California  dug  uniformly  some  ounces 
of  gold  per  day,  for  some  days,  when,  becoming  one  of 
a  company,  consisting  in  all  of  as  many  miners  as  he  had 
worked  days  alone,  he  received  his  share  of  500  ounces, 
when  the  entire  company  changing  to  as  many  miners  as 
he  had  dug  ounces  per  day  alone,  he  received  his  share  of 
342  ff  |  ounces.     After  giving  just  this  information  to  a 
speculator,  the  latter  agreed  to  pay  him  for  185  ounces  of 
gold  for  his  all.     Construing  these  conditions  most  favora 
bly  to  the  speculator,  can  he  gain  any  thing  ? 

Ans.  He  must  lose  the  value  of  18^  ounces. 

6.  Some  farmers  bartered  animals  :  in  exchange  for  5 
heifers,  Smith  gave  Jones  7  sheep  and  3  dollars  ;    Johnson 
gave  Taylor  4  heifers  and  2  dollars  for  6  sheep,  and  Simp 
son  gave  Thomson  a  heifer  for  a  sheep.     After  these  trans 
actions  an  army  agent  purchased  all  these  same  animals  as 
of  an  approved  and  standard  value,  each  kind  ;  which  gave 
rise  to  conversation  among  them  as  to  who  gained  in  their 
mutual  trades.     The  three  who  gained,  each  agree  to  mul 
tiply  the  number  of  dollars  gained,  by  itself,  add  the  pro 
ducts  together  as  so  many  dollars,  and  give  this  sum  of 
dollars  to  the  person  who  would  tell  them  what  it  would 
be  when  it  was  the  smallest  it  could  be  for  any  value  of 
those  animals,  as  that  standard  value  ;  and  required  that 
sum. 

If  x  =  number  dollars  value  of  a  sheep, 

and  y  —  number  dollars  value  of  a  heifer, 

and  z  =  that  sum  of  money  required  ; 

then  z—(lx  —  5?/  +  3)2+(6^  —  4y  —  2)  2  -+-(«  —  y)2, 


or,  all  the  signs  of  the  quantities  within  the  parentheses 
may   be  changed,    since    it   will  not  affect  2,    and    since 


140  DIFFERENTIAL   CALCULUS. 

we   cannot    presume    that    x  is    either    greater    or  less 
than  y. 

Ans.  A  heifer,  5£  dollars  ;  a  sheep,  3£;  Smith,  Johnson, 
and  Simpson  gained  1  §•  dollars  each  ;  sum  required 
8  £  dollars. 

The  following  problems,  (a.)  and  (5.),  contain  each  but 
one  independent  variable : 

7.  («.)  A  carriage  wheel,  which  was  in  circumference  5 
times  the  length  of  step  of  a  certain  pedestrian,  and  1  foot 
more,  the  length  of  that  step  being  2£  feet,  ran  once  over 
a  route  10  times  as  long  as  that  between  Dock  Square  in 
Boston  and  a  certain  station  A,  and  then  ran  1000  feet  more. 
A  second  wheel,  of  such  size  that  it  would  revolve  500  times 
in  going  once  between  Dock  Square  and  station  A,  ran 
once  between  Bowdoin  Square  and  station  ./?,  a  distance 
equal  to  1200  of  those  steps.  Required  the  distance  from 
Dock  Square  to  A,  when  2,  the  sum  of  all  the  revolutions 
of  the  two  wheels,  is  a  minimum  or  maximum. 

Ans.  Distance  1679  feet,  and  z  is  a  minimum,  it  being 
then  2501.4. 

(b.)  A  carriage  wheel,  which  was  in  circumference  5 
times  the  length  of  step  of  a  certain  pedestrian,  and  1  foot 
more,  ran  once  over  a  route  10  times  as  long  as  that  be 
tween  Dock  Square  and  a  certain  station  A  (which  was 
a  distance  of  1G79  feet),  and  then  ran  1000  feet  more.  A 
second  wheel,  of  such  size  that  it  would  revolve  500  times  in 
going  once  between  Dock  Square  and  station  A,  ran  once 
between  Bowdoin  Square  and  station  J?,  a  distance  equal 
to  1200  of  those  steps.  Required  the  length  of  that  step 
when  the  sum  2,  of  all  the  revolutions  of  the  two  wheels,  is 
a  minimum. 

Ans.  Step  3.28  feet;  now  z  is  a  minimum,  at  2500.8. 


MAXIMA,    ETC.,    WITH   TWO    VARIABLES.  141 

The  following  problem  (c.)  is  the  same  as  (a.)  and  (b.) 
preceding,  except  that  it  combines  in  one  (function  of  x 
and  y)  =  2,  each  of  the  same  variables  as  in  (a.)  and  (b.)  ; 
thus  making  two  independent  variables  : 

(c.)  A  carriage  wheel,  which  was  in  circumference  5 
times  the  length  of  step  of  a  certain  pedestrian,  and  1  foot 
more,  ran  once  over  a  route  10  times  as  long  as  that  between 
Dock  Square  and  a  certain  station  A,  and  then  ran  1000 
feet  more.  A  second  wheel,  of  such  size  that  it  would 
revolve  500  times  in  going  once  between  Dock  Square  and 
station  ^4,  ran  once  between  Bowdoin  Square  and  station 
j#,  a  distance  equal  to  1200  of  those  steps.  Required  the 
distance  from  Dock  Square  to* -4,  and  the  length  of  that 
step  when  z,  the  sum  of  all  the  revolutions  of  the  two 
wheels,  is  a  maximum  or  minimum. 

8.  (a.)  A  certain  perpendicular  flag-staff,  129  feet  high, 
stands  on  a  level  plain ;  a  stake  is  driven  into  the  ground 
to  mark  a  point  60  feet  to  the  south  of  the  base  of  that 
staff,  which  point  is  joined  with  the  top  of  the  staff  by  a 
straight  cord.  Another  such  flag-staff,  97  feet  high,  stands 
82  feet  to  the  westward  of  the  first,  and  its  top  is  joined 
by  a  cord  to  a  point  at  the  surface  of  the  ground  marked 
by  a  stake  40  feet  to  the  east  of  the  first  stake.  Required 
to  determine  the  nearest  distance  between  one  cord  and 
the  other,  either  produced  indefinitely  if  necessary,  which 
might  be  the  case  in  a  generalization  of  the  conditions. 

It  will  be  useful,  in  the  solution  of  the  above  problem,  to 
conceive  three  arbitrary  planes  cutting  each  other  at  right 
angles,  to  each  of  which  any  point  in  either  cord  may  be 
referred  by  perpendicular  measurement ;  through  the  me 
dium  of  right-angled  triangles,  an  expression  may  be  found 
for  a  perpendicular  Jine  from  one  cord  to  the  other ;  this 
line  must  be  a  minimum.  By  means  of  a  solid  diagram, 


142  DIFFERENTIAL   CALCULUS. 

constructed  of  pasteboards  for  planes  and  threads  for  lines, 
there  is  much  simplicity  in  the  solution. 

(b.)  Required  to  determine  the  diameter  of  the  smallest 
sphere  to  which  the  above  two  lines  are  tangent,  and 
whether  this  diameter  is  the  line  required  in  the  foregoing 
question. 

The  following  problems  trespass  upon  the  rule  hitherto 
adhered  to,  not  to  propose  geometrical  and  trigonometrical 
problems,  except  the  most  elementary. 

9.  A  sentinel  has  received  orders  from  his  commanding 
officer  to  visit  in  succession  three  important  posts,  A,  _/?, 
and  (7,  and  return  from  each  visit  of  each  to  his  camp. 
The  post  A  is  90  rods  from  7?,  13  23  rods  from  (7,  and  C 
72  rods  from  A.     But  he  may  place  his  camp  where  he 
pleases.     Required  the  distance  of  his  camp  severally  from 
A,  B,  and  (7,  when  a  round  of  visits  is  made  with  the  fewest 
steps,  and  consequently  any  number  of  rounds,  the  ground 
supposed  level.  

The  following  problem  is  to  be  considered  general,  with 
reference  to  Sections  XII.,  XVI.,  and  XVIII. 

10.  A  plain  is  level,  and  the  sight  over  it  is  unobstructed 
by  objects  :  on  it  is  a  circular  course  80  rods  in  diameter. 
A  procession  was  once  seen  marching  round  this  course,  in 
which  was  a  person  carrying  a  banner  4  feet  square,  and 
holding  it  perpendicularly,  with  its  centre  9  feet  above  the 
plain,  and  40  rods,  horizontally  measured,  from  the  centre 
of  the  course.     The  banner  was  constantly  held  with  its 
plane  agreeing  with  the  radius  of  this  course,  and  hence 
invisible  to  a  person  at  the  centre  of  the  course.     But  out 
side  there  was  a  stationary  observer,  so  situated  that  his 
eye  was  70  rods  from  the  centre  of  the  course,  and  9  feet 
above  the  plain.     As  the  banner  was  thus  carried  entirely 


ELEMENTARY   DEMONSTRATIONS.  143 

round  this  course,  its  two  sides  being  successively  exhibited 
to  that  observer,  it  is  probable  that  there"  were  two  loca 
tions,  one  toward  the  right,  the  other  toward  the  left  of  the 
observer,  where  that  banner  appeared  the  largest  object,  as 
when  it  should  be  projected,  when  any  where  on  its  circuit, 
to  form  a  portion  of  the  surface  of  a  sphere,  of  which  the 
observer's  eye  is  at  the  centre,  such  surface  being  assumed 
at  any  distance  whatever.  Required  the  distance  in  rods 
from  the  eye  to  centre  of  banner,  when  the  banner  ap 
peared  largest. 

The  planet  Yenus  gives  her  maximum  light  to  the  earth 
on  conditions  not  much  unlike  the  above. 


SECTION  XIX. 

DEMONSTRATION,  OF  THE  GENERAL  FORM  OF  THE 
DEVELOPMENT  OF  /  (x  -f-  A),  AND,  OF  THE  DIFFER 
ENTIATION  OF  CERTAIN  FUNCTIONS. 

158,  We  have  deferred  to  the  present  section  a  very 
elementary  and  important  demonstration  in  regard  to  the 
form  of  the  development  of  any  function  of  x  whatever, 
when  x  becomes  x  -\-  h,  and  our  earlier  endeavors  to  illus 
trate  the  nature  and  rules  of  differentiation  were  at  a  dis 
advantage  on  account  of  the  omission. 

Let  fx  be  any  function  of  x,  and  when  x  becomes 
x  -|-  A,  f  (x  -(-  A)  will  have  a  general  development  of  the 
form 


f(x  +  h)  =fx  +  Ah-{-gh*+Ch*+,  etc., 

in  which  A,  .Z?,  (7,  etc.,  are  coefficients  containing  x  and 
constants,  and   each  may  evidently  be  an  aggregate  of 


144  DIFFERENTIAL   CALCULUS. 

certain  sub-terms,  etc.  The  form  is  intended  to  show  how 
h  appears  in  the  development. 

It  is  useful  to  prove  the  foregoing  general  development 
without  reference  to  the  binomial  or  any  other  theorem. 

One  of  the  terms  of  the  general  development  must  befx, 
in  which  h  in  no  respect  exists  as  a  factor  or  otherwise, 
because  when  h  =  0,  and/"  (x  -|-  h)  becomes  fx,  the  de 
velopment  ought  to  reduce  to 


Nor  can  there  be  any  term  but  fx  in  the  development 
which  does  not  contain  A  as  a  factor. 

None  of  the  indexes  of  h  can  be  negative  ;  because  if  h 
have  a  negative  index,  h  may  be  made  to  appear  as  a 
denominator  of  A,  JB,  or  (7,  etc.,  with  that  index  positive. 
Such  term  would,  therefore,  become  infinite  when  h  =  0, 
but  when  h  =.  0  the  term  itself  ought  to  become  0,  because 
the  condition  of  the  equation  becomes  f  x  =fx,  and^ic 
is  not  necessarily  infinite,  nor  has  it  any  restricted  value, 
nor  any  value,  therefore,  that  ought  to  be  restricted  in  the 
development.  Nor  can  there  be  in  the  series  two  terms, 
each  infinite  and  with  opposite  signs,  because  they  would 
not  be  equal  infinites  unless  they  should  be  rendered  so 
by  the  vanishing  of  h  at  like  powers  or  like  rates,  and  in 
such  case  such  two  terms  become  one  in  the  series. 

None  of  the  exponents  of  h  can  be  fractional,  because 
of  one  factor  of  such  term,  a  root  is  indicated  to  be  taken. 
Now  all  these  terms  of  the  general  development  are  sup 
posed  to  be  numerical;  and  the  roots  only  of  particular 
numerical  quantities  are  rational,  such  as  1,  £,  ^  4,  16,  8,  27, 
etc.  ;  the  roots  of  intermediate  quantities  may  be  irrational 
or  inexpressible  in  number.  Therefore  the  roots  of  numeri 
cal  quantities  in  general  are  irrational.  This  truth  is  not 
affected  by  the  consideration  that  the  roots  of  powers  are 


ELEMENTARY   DEMONSTRATIONS.  145 

indicated  by  fractional  exponents.  Now  if  a  root  of  h  be 
irrational,  the  term  which  contains  it  is  irrational.  Not 
withstanding  what  the  character  of  the  other  terms  of  the 
development  may  be,  f  (&  -\-  h)  being  equated  with  an 
expression,  one  of  the  terms  of  which  is  irrational,  is  itself 
irrational  in  general  values  /  hence  a  restriction  is  imposed 
upon  the  values  of  the  development  of  /  (x  -f-  h),  and 
upon/'ic,  and  such  development  is  not  general. 

The  indexes  of  h  in  the  successive  terms,  are  the  natural 
series  of  entire  and  positive  numbers,  1,  2,  3,  etc.  ;  for  if  A 
be  the  coefficient  of  h  at  the  lowest  power  or  a,  we  may 
write  the  development  thus  : 


a+,  etc.,)  h  a  ; 
but  which  for  simplicity  we  will  write  thus  : 


A  +  P 


wherefore  if  a  is  not  a  unit,  we  have/  (x  -\-  h)  —  fx  ren 
dered  irrational,  and  h  itself  irrational,  and  consequently 
each  at  restricted  values,  which  are  opposed  to  the  hypothe 
sis  ;  therefore  a  is  unity. 
Since  a  is  unity,  we  have 


or,  rather, 


Again,  as  before, 

f(x+h)—  fx—Ahi= 
13 


146  DIFFERENTIAL   CALCULUS. 

but  which,  for  simplicity,  we  will  put 

/  (x  +  A)  —  fx  —  Ahi=(£+ 


i 

=  A; 


which  imposes  restrictions  against  hypothesis  unless  5  =  2, 
for  when  5  =  2,  we  find  g^ri  =  1.  Now,  by  the  continua 
tion  of  this  course  of  reasoning,  we  may  show  c  =  3,  etc. 
Therefore  the  indexes,  etc. 

If  h  be  negative  as  in  f  (x  —  A),  it  is  evident  that  the 
general  development  is  the  same  in  form,  except  that  the 
terms  having  h  with  indexes  odd,  will  be  negative. 

The  coefficients  A,  B,  (7,  etc.,  in  the  general  develop 
ment,  are  evidently  functions  of  ce,  but  are  not  as  yet,  ex 
cept  A,  differential  coefficients.  (Art.  99.) 

This  development  being  general,  holds  for  such  particular 
cases  as  render  one  or  more  of  its  terms  imaginary. 

Although  authors  are  very  reserved,  and  some  of  them 
entirely  silent,  respecting  the  possible  value  of  h  in  this 
development,  it  is  plain  that  it  must  be  an  infinitesimal,  or 
indefinitely  small.  If  in  the  formula,  2  A  be  substituted 
for  A,  it  becomes 

f  (x  _L.  2  A)  =fx  +  2  A  h  +  4  B  A2  -f  8  C  A3  +,  etc., 

which  shows  that  the  coefficients  A,  J5,  (7,  etc.,  are  affected 
by,  and  rendered  dependent  upon,  a  change  of  the  value 
of  A. 

159.  By  transposing  the  first  term  in  the  general  de 
velopment  of  f  (x  -f-  A),  we  have 

/(a-  4.  A)  —fx  =  A  A  +  7?  A2-)-  <7A3+,  etc. 
Now  the  first  member  of  this  equation  is  the  amount  that 


ELEMENTARY   DEMONSTRATIONS.  147 

f  x  changes  by  virtue  of  A  added  to  the  variable  x.  Divid 
ing  by  A,  the  increment  of  x,  we  have  the  fractional  or 
proper  form  of  expressing  the  ratio  of  the  change  of  value 
of  the  function  to  that  of  the  variable,  that  is, 


,  etc. 


Now  when  h  —  0,  the  numerator  may  take  the  designa 
tion  dfx  (or  if  fx  =  ?/),  of  d  y,  and  the  denominator  h 
must  take  the  designation  d  x. 


"dx~ 

which  we  may  enunciate  thus  :  the  coefficient  of  the  second 
term  of  the  general  development  of  f  (x  -j-  h)  is  the  dif. 
coef.  derived  from  f  x.  (Art.  68.) 

160.  The  form  of  the  general  development  of/  (x  -f-  A) 
furnishes  the  means  of  a  formal  demonstration  of  the 
method  of  differentiating  the  product  of  two  or  more  func 
tions  of  the  same  variable. 

Let  y  and  z  be  functions  of  x  in  the  expression 

u  =  a  y  z. 

By  changing  x  into  x  -\-  A,  the  function  y  becomes  (desig 
nating  by  y'  the  new  value  of  y) 


,  etc.,  (1.). 

and  the  function  z  becomes 

z1  =  z  +  A1  h  +  Br  A  2  _j_  C<  A3  +,  etc.         (2.). 
Hence,  when  A  =  0,  we  have  from  (1.), 

y'  —  y  _  dy  __ 

~h~—  dx~  ^; 


148  DIFFERENTIAL   CALCULUS, 

and  from  (2.), 

z  '  —  z         d  z 

-        ___     A  I 

h        ~  dx~ 

Designating  by  u'  the  new  value  of  u  received  in  conse 
quence  of  the  change  of  y  and  z,  and  multiplying  the 
product  of  (1.)  and  (2.)  by  a,  we  have 

ur  =  a  y  z  -\-  a  (A  z  -\-  A1  y)  h  -|-,  etc., 

==ay«  +  a(^s  +  ^y)  h +,  etc.; 
therefore  a  ( —  z  -\ y  )  being  the  coef.  of  the  second 

\d  x  d  x      J 

term  of  the  development  of  u',  we  have 

d  u  d  y  d  z 

—  =  a  z  — -  -\-  aii  —  : 

dx  dx     '         y  dx 

.-.du  =  azdy-\-aydz.  (3.). 

Hence,  to  differentiate  the  product  of  two  functions  of 
the  same  variable,  we  must  multiply  each  by  the  differential 
of  the  other,  and  add  the  results. 

161t  It  will  be  easy  now  to  express  the  differential  of  a 
product  of  three  functions  of  the  same  variable.  Let 

u  =  w  y  z 

be  a  product  of  three  functions  of  x ;  then,  putting  v  for  w  y, 
the  expression  is 

U  =2  V  Z\ 

hence  by  (3.), 

du  —  z  dv  -f-  v  dz, 

but  v  =  w  y ;  therefore  by  (3.), 


ELEMENTARY   DEMONSTRATIONS.  149 

consequently  by  substitution 


and  it  is  plain  that  in  this  way  the  differential  may  be  found, 
be  the  factors  ever  so  many  ;  so  that  generally,  to  differen 
tiate  a  product  of  several  functions  of  the  same  variable,  we 
must  multiply  the  differential  of  each  factor  by  the  product 
of  all  the  other  factors,  and  add  the  results.  (Arts.  74,  75.) 

163t  If  it  be  required  to  differentiate  an  expression  con 
sisting  of  several  functions  of  the  same  variable  combined 
by  addition  or  subtraction,  it  will  be  necessary  merely  to 
differentiate  each  separately,  and  to  connect  together  the 
results  by  their  respective  signs.  For  let  the  expression  be 

u  =  aw-{-by-\-cz  -(-,  etc., 

in  which  w,  y,  and  z  are  functions  of  x.  Then,  changing  x 
into  x  -\-  h,  and  developing, 

w  becomes  w  -\-  A  h     -\-  It  h*     +>  etc., 

y        «        y  +  A'h  +  B'  h*  +,  etc., 

z       «         z  +  A"  h  +  £"  h*  +,  etc., 
...  w        «        u-\-(aA-{-bA'-\-cA"-{-,etc.),h-\-,etc., 

.'.du  =  aAdx-\-b  A'  d  x  -\-  c  A"  d  x  -\-,  etc. 
But    A  dx  =  dw,  A  dx  =  d  y,  A"  dx  =  dz,  etc.  ; 
-bdy-\-cdz~\-,  etc.  ; 


that  is,  the  differential  of  the  sum  of  any  number  of  func 
tions  is  equal  to  the  sum  of  their  respective  differentials. 
(Art.  76.) 

13* 


150  DIFFERENTIAL  CALCULUS. 

SECTION  XX. 

MACLAURIN'S  THEOREM,  AND  ITS  APPLICATION. 

103.  This  theorem  gives  a  general  mode  of  developing, 
expanding,  or  changing  the  form  of,  some  algebraic  (and 
other)  expressions  by  a  series  arranged  with  reference  to 
the  positive  ascending  powers  of  any  one  specific  quantity 
in  them,  which  may  be  assumed  for  the  purpose. 

A  function  having  a  single  variable  is  such  an  expres 
sion,  and  the  variable  may  be  selected  as  the  specific 
quantity  in  question.  Since  only  the  form  is  changed,  the 
value  of  the  expression,  or  of  the  function  (if  it  have  a 
value)  must  remain  unchanged,  except  in  marked  excep 
tional  cases.  If  a  function  has  no  specific  value,  the  new 
form  of  it  produced  by  this  theorem  must  have  the  same 
range  of  values,  if  the  values  are  real,  as  the  original  func 
tion.  Sometimes  the  odd,  sometimes  the  even  powers  of 
the  specific  quantity  become  eliminated  from  the  series, 
because  the  coefficients  of  such  terms  must  respectively 
equal  zero.  If  x  be  the  quantity,  or  represent  the  position 
of  the  quantity  according  to  the  ascending  powers  of  which 
the  series  is  to  be  formed,  then  the  expression  being  called 
a  function  of  #,  and  A,  _Z?,  C",  etc.,  being  indeterminate 
coefficients,  successively,  of  the  powers  of  a*,  we  have 

y—  A  +  J3x  +  Cx*  +  Dx*  +  Ex*  -f,  etc.; 
.-.  —  =      £  +  2  Ox  +  3  D  x*  +  4  E  x*  +,  etc.  ; 

d  x 


-=:  2  (7+2.3  J)x  +  3.4^^2+,  etc.; 

d  x2 

?!?=  +2.3J>  +  2.3.4^ic+,  etc.  ; 

d  x3 


MACLAUBIN'S  THEOREM.  151 


.-.         =  2.8.4^,  etc. 

dx* 

Now,  if,  by  making  x  =  0,  we  select  the  particular  val 
ues  respectively  for  these  coefficients,  and  the  function, 
which  are  not  affected  by  the  value  of  x,  and  place  them  in 
parentheses  to  denote  this,  we  have 


1.2.3.4 

Substituting  these  expressions  for  A,  £,   (7,  etc.,  the 
series  becomes 


eto, 


which  is  Maclaurin's  Theorem. 

Although,  to  derive  these  coefficients,  x  was  made  equal 
to  zero,  yet  by  hypothesis  they  are  such  as  the  value  of  x 
cannot  affect  ;  therefore,  in  the  theorem,  x  may  be  restored 
to  any  value  consistent  with  the  function. 

Otherwise  : 
Taylor's  Theorem  being, 


,    dFx   h    ,    d*Fx   h*      ,    d*Fx      h* 

Fx  +  -  ---  -----  ----  K  etc.  ; 

dx      1    '      dx*     1.2    '      dx*     1.2.3    ' 


152  DIFFERENTIAL   CALCULUS. 

if  x  =  0,  it  becomes 


where  the  parentheses  are  used  to  intimate  that  x  has  this 
restricted  value,  x  =  0.  These  differential  coefficients,  in 
parentheses,  are  constant,  for  in  the  actual  coefficients  of  a 
particular  case  applied  under  this  general  notation,  x  will 
not  be  found.  Therefore,  h  is  no  longer  limited  to  the 
value  0,  but  may  be  of  any  greatness  ;  and  since  x  has  dis 
appeared,  we  may  revive  it  in  h.  We  then  have,  calling 
the  original  F  x  =  y  and  (F  x)  =  (y), 


which  is  Maclaurin's  Theorem. 

1 


1.   Required  the  development  of  y  =  F  x,  viz., 


dx  (a+  x)z         \dx 

_  2        ^ 

~      ""    a  +  z)3  "  \dx3 


_  2.3 

rfa;3  ~      ^  (o  +  *)  *  * 


1  1  XX-  X 

"  a  +  x          a         a2     '     a3         a4 

2.   Required  y  =  (#2  -|-  x2)  *  in  a  series. 
s.y  =  &  +  -*—      "4 

«r  r>  i 


MACLAURIN'S  THEOREM.  153 

4.  It  is  required  to  express  in  a  series  of  the  ascending 
positive  powers  of  x  this  function  of  ic,  viz., . 

a  a          a  a  a 

Ans. = "2^^ — ~sx ic34~>  etc. 

5.  It  is  required  to  develop  the  algebraic   expression 
V  («2  +  #2)  in  a  series  with  reference  to  the  increasing 
positive  powers  of  b. 

b*  b*  be 

Ans.  V  («2  +  #2)  =  a  H 1 >  etc. 

'2a         8  a3     '     16  a5 

6.  Change  the  form  of as  a  function  of  x. 

h  — a  x 

Ans. 


b  —  a  x 
c  c      /  ax         a*x*         a3  x3 


b  —  ax         b 


7.  Develop,    if    possible,    y  =  a  x4    by    Maclaurin's 
Theorem. 

y  —  ax\  .-.  (y)  =  0  ; 


/.  a  *4  =  0       0       0       0       24a  --  +  0, 


154  DIFFERENTIAL   CALCULUS. 

Whenco  it  appears  that  in  this  case  the  theorem  does 
not  fail  in  truthfulness,  but  in  utility.  The  development 
is  not  a  series. 

8.   Develop  into  a  series  y  •=. 


l  —  x 

Ans.  y  =  1  +  x  -\-  x2  -|-  x3  -\-  x4  -{- ,  etc. 

Whence  it  appears  that  the  theorem  fails  to  give  an 
equivalent  substitute  for  y  when  1  <^  sc,  because  the  sub 
stitute  becomes  infinite. 

It  may  have  been  noticed  that  all  the  results  obtained 
in  the  foregoing  examples  might  also  be  obtained  by  ordi 
nary  algebraic  division  or  extraction  of  roots,  or  at  least  by 
the  Binomial  Theorem. 

But  we  ordinarily  do  not  have  an  algebraic  demonstra 
tion  of  this  theorem  so  general  as  to  embrace  develop 
ments  in  case  the  indexes  are  fractional,  denoting  both  a 
power  and  a  root,  as  f.  Although  this  theorem  embraces 
the  binomial,  it  is,  therefore,  more  general.  It  is  the  foun 
dation  of  other  theorems  like  Lagrange's,  and  appears  in 
dispensable  in  the  higher  Calculus,  and  in  trigonometrical 
analysis. 

9.   Let  it  be  required  to  develop 

s+l 


10.   Let  it  be  required  to  develop 

a  x  +  V~# 

164,  Binomial  Theorem.  If  the  expression  (a  -|-  x) n 
be  developed  by  Maclatirin's  Theorem,  the  result  exhibits 
the  Binomial  Theorem. 


BINOMIAL  THEOREM.  155 

11.  Let  it  be  required  to  develop  Fx,  or  y  =  (a  -\-  x)  n, 
according  to  the  ascending  positive  integral  powers  of  x, 
or  by  Maclaurin's  Theorem  : 

We  have         y  =  (a  -f-  x)  n,  /.  (y)  =  an, 


2)  (a 


=»(»-!)(»  -2)  a"-'; 


.-.       n=n(n  —  l)(n—  2)  (n—  3)  ...(w  —  n)  (a-f  »)n~w. 
Now,  y  being  ^^  with  reference  to  Maclaurin's  Theorem, 

/rfyx  /dFx\     /d*y\  /d*Fx^ 

we  have  (  --   )  =  (  ---    ;  I  -  -  )  =     -      -  ),  etc.;  hence, 

\dx)     \  dx  )  \dx*)       \  dxz  r 


substituting  the  equivalents  of  these  as  found,  we  have 

„  n\  n-i  n(n—\)     n_2 

—  y        V     T~ x)  2 

n  (n  _  i)  (n  -  2)      M_S 
1  '  \Jj  t)C>        \ —  *  C LC«« 

2.3 

which  is  the  Binomial  Theorem,  and  n  may  be  a  whole 
number  or  a  fraction,  or  be  negative  or  irrational.  It  will 
be  observed  that  when  n  is  a  whole  number,  since  n  —  n 


156  DIFFERENTIAL   CALCULUS. 

becomes  a  factor  in  the  nth  dif.  coef.,  it  renders  it  =  0,  and 
the  series  must  terminate. 

In  case  the  index  n  should  not  be  a  whole  number,  but 

should  be  a  fraction  like  —  when  in  its  simplest  form,  then 
the  differences  -  —  1,  -  —  2,  ...  -  —  n,  etc.,  by  the*  suc 

cessive  subtractions  of  the  integral  numbers  1,  2,  3  ...  ft, 
could  never  be  reduced  to  0,  but  would  pass  over  0  in 
going  from  -{-  to  —  values  ;  and  the  series  would  never 
terminate. 

165,  But  the  mode  of  developing,  by  Maclaurin's  Theo 
rem,  a  function  of  a*,  is  applicable  as  well  when  the  func 
tion  is  implicit.  It  may  be  observed  that,  when  x  is 
made  =  0,  y  being  dependent  will  take  some  value  in 
constants  corresponding  to  x  =  0,  which  value  is  to  be 
substituted  for  y. 

12.  Let  it  be  required  to  develop  y  according  to  the 
ascending  powers  of  x  in 


m     fdy\   ___J_. 

~~  ~ 


~d~x         32  —  3'   "   \dx 


dx*        (3y2  — 3)a 


dx* 


.    fg!^  —  6X3' (*X  &)__!!_.• 
'  Vrfx3/  ~  34  34  ' 

a;          a-3          a:5 

•••^i  +  3T+J7  +  'ete- 


BINOMIAL   THEOREM.  157 

It  may  be  observed  that  the  differential  of  the  numera 
tor  6  y  —  ,  this  being  a  product  of  the  two  variables  y  and 

dx 

dy^dx  as  wfcll  as  —  ,  being  constant,  and  d^  y  being 
the  differential  of  dy,  is 


which  terms,  as  well  as  the  others  of  the  new  numerator 
which  is  being  formed,  are  to  be  divided  by  dx>  as  well 
as  the  other  member  of  the  equation,  which  is  now 

d3  y 

—  ;  hence  we  have,  in  the  third  dif.  coef.,  as  a  factor 

a  x2 


dx    dx  d 

Since  y  =  0  when  x  =  0,  all  terms  containing  y  as  a  fac 
tor  become  eliminated  in  finding  those  dif.  coefs.  within 
brackets. 

13.  Let  it  be  required  to  develop  y  in  b  y  3  —  xy  =  b, 
according  to  the  ascending  powers  of  x. 

Ans.      =  l_-fiLaetO. 


14.   Let  it  be  required  to  develop  y  in  y3x  —  a3  (y-\-x) 
=  0,  according  to  the  powers  of  x. 

Ans.  y  =  —  x  —  -  ---  -  --  etc. 

a  3  a6 

166,  It  should  not  escape  notice  that  here  is  shown  an 
achievement,  by  the  analysis  of  the  calculus,  beyond  the 
ordinary  direct  resources  of  algebra,  in  relation  to  the 
resolution  of  equations,  and  is  particularly  available  when 
ever  the  series  generated  in  the  manner  shown,  is  so  con 
verging  that  the  terms  of  it  may  be  readily  summed  with- 
14 


158  DIFFERENTIAL   CALCULUS. 

out  tediousness,  as  when  the  denominators,  when  there 
are  any,  are  large  relatively  to  the  numerators,  or  to  &, 
which  may  always  be  in  a  numerator,  when  the  terms  are 
fractional. 

Nor  should  it  escape  notice  what  may  be  accomplished 
by  the  transformation  of  x  into  y,  or  y  into  ic,  or  of  any 
known  constant  into  x  or  y,  or  by  creating  x  at  pleasure, 
with  the  substitution  of  it  for  any  term  or  quantity,  for 
the  sake  of  the  development  ;  and  of  the  means  of  equating 
any  unknown  quantity  situated  as  y,  with  others  which 
are  presumed  to  be  known. 

It  is  useful  to  observe  that  these  general  analyses  never 
fail  to  embrace  the  truth  of  quite  elementary  conditions, 
even  when  a  series  may  fail  to  have  place.  For,  — 

15.  Let  it  be  required  to  develop  y  in  y  3  —  x3  =  Q, 
according  to  the  powers  of  x. 

We  have  y  =  x,  .-.  (y)  =.  0  ; 

dy_  _  3;r2         /dy\    _      ^ 
dx         3</2  "  \dx)  ~ 
-3x* 


dx 


dx* 


dx* 

.'.  y  i=0  +  ^  +  0+,  etc.; 

3z8 
where  we  are  obliged  to  remark  that  the  foregoing  —^  ? 

as  well  as  the  second  dif.  coef.,  in  its  general  form,  are 
reduced,  respectively,  to  1  and  0,  from  expressions  each 
virtually  -  when  x  =  0,  by  principles  that  will  be  fully 


BINOMIAL  THEOREM.  159 

demonstrated  in  the  following  section.    At  present  it  is 
sufficient  to  say  that 


because  both  numerator  and  denominator  become  alike 
when  x  =  0,  and  the  second  dif.  coef.,  that  is, 

6a  X  3  y2  —  3  a2  X  6y  z=  0 

when  x  and  y  are  any  how  alike,  without  notice  of  the 
circumstance  that  their  respective  values  are  0,  which 
casually  renders  9  y  4  =  0. 

167.  Maclaurin's  Theorem  fails  to  give  a  true  develop 
ment  of  all  functious  of  a?,  of  which  any  dif.  coef.  becomes 
infinite  when  x  =  0.  We  know  nothing  of  a  development 
by  it,  when  x  should  be  restricted  to  the  value  0,  and  of 
which  any  dif.  coef.  becomes  infinite.  And  since  in  the 
theorem,  x  in  the  position  of  all  its  ascending  powers  is 
not,  by  the  nature  of  the  theorem,  to  be  restricted  to  any 
value,  when  0  <  x  any  such  term,  and,  as  will  be  seen  by 
a  few  examples,  all  succeeding  terms  become  infinite. 
Such,  therefore,  cannot  be  a  development  of  a  function 
which  is  not  necessarily  infinite.  Thus,  if 

y  —  «*, 

dy         1 
-  =  —  i  =  co  when  x  =.  0, 

dx         2*5 

-  =  --  —  r  =  —  co  when  x  =  0  . 
d  x*  4*4 

So,  also,  with  a  x  J,  (a  x  —  a;2)  1,  b  x  i,  etc.  But  the  infi 
nite  dif.  coef.  might  be  deferred  to  the  5th,  6th,  or  the  nth, 
for  the  obvious  reason  that  several  successive  subtractions 
of  unity  from  an  improper  fraction  may  be  necessary  be 
fore  the  remainder  becomes  negative. 


160  DIFFERENTIAL   CALCULUS. 

168.  In  Arts.  151,  153,  we  have  given  the  .development 
of  a  function  of  two  independent  variables  by  Taylor's 
Theorem.  In  the  same  manner  in  which  we  have  derived 
Maclaurin's  Theorem  from  Taylor's  as  for  one  variable, 
we  may  derive  the  development  of  a  function  of  two  inde 
pendent  variables  by  Maclaurin's.  If  in  that  development 
(Art.  152)  we  suppose  x  and  y  each  =  0,  the  develop 
ment  will  become  that  of  F  (h>  k)  according  to  the  powers 
of  h  and  Jc,  or  substituting  in  that  development  x  for  h  and 
y  for  k,  since  these  may  now  have  any  value,  we  have 


16.   Let  it  be  required  to  develop  z  according  to  the 
powers  of  x  and  y  in 

z  =  ax*  (b  —  y3)  —  y2  (z2-i-  c)2. 


SECTION  XXL 

DETERMINATION    OF    THE    VALUE    OF    VANISHING 
FRACTIONS. 

109.  We  have,  on  several  occasions,  compared  the  rates 
of  change  of  value  of  two  functions  of  the  same  variable, 
for  some  particular  value  of  the  variable.  In  such  case,  the 
variable  may  not  only  be  expressed  as  x  in  each  function, 
but  by  hypothesis  is  to  be  the  same  #,  and  therefore  is  to 
have  a  common  value  in  each  function. 

When  such  two  functions  become,  respectively,  the  nu- 


VALUE   OP   VANISHING   FRACTIONS.  161 

merator  and  denominator  of  a  fraction,  the  case  may  hap 
pen  when,  on  the  variable  taking  a  particular  value,  the 

fraction  reduces  to  the  form  - ;  such  is  called  a  vanishing 

fraction.  This  value  is  indeterminate  in  the  abstract,  but 
determinate  when  we  know  its  origin.  The  value  of  a 
vanishing  fraction  does  not  necessarily  vanish.  The  nu 
merator  and  denominator  vanish  severally  and  independ 
ently,  or  by  independent  rates  of  change. 

1.  On  an  occasion  it  cost  a  man  75  cents  per  mile  to 
travel ;  however,  of  the  whole  number  of  miles  travelled, 
42  were  without  cost.     If  a  sum  like  that  expended  in  this 
travelling,  should  be  expended  in  the  purchase  of  25  times 
as  many  pounds  of  the  commodity  C  as  he  had  travelled 
miles  with  cost,  what  would  have  been  its  price  per  pound, 
whatever  the  number  of  miles  travelled  with  cost  might 
have  been,  even  if  it  had  been  the  least  conceivable  in  a 
fraction  ? 

Let  x  =  number  of  miles  travelled  in  all ;  then  the  price 
per  pound  of  C  will  be  represented  thus  : 

(x  -  42)  75 

—  =  3  cents. 
(x  —  42)  25 

Here  it  is  evident  that,  in  case  x  =  42,  the  fraction  re 
duces  to  -;  but  its  value  appears  to  be  3  nevertheless.    If 

we  watch  the  relative  values  of  the  numerator  and  denom 
inator  while,  by  a  variation  of  a*,  they  are  becoming  ex 
ceedingly  small,  it  is  quite  evident  that  nothing  disturbs 
the  ratio  of  their  values. 

2.  A  courier  travelled  15  or  a  hours,  at  15  or  a  miles 
per  hour,  in  a  continuous  course,  when  he  travelled,  in 
return,  just  as  many  hours  as  miles  per  hour ;  we  need 
not  say,  as  yet,  whether  or  not  he  had  accomplished  just 
his  return,  but  a  messenger  was  ready  to,  or  did  proceed 

14* 


162  DIFFERENTIAL   CALCULU3 

to  meet  him  at  a  rate  per  hour  equal  to  the  excess  of  the 
rate  per  hour  of  the  courier's  set-out,  above  the  rate  of  his 
return,  per  hour.  Required  to  determine  the  number  of 
hours  necessary  for  the  messenger's  travel,  although  the 
distance  necessary  for  him  to  travel  were  the  shortest 
conceivable. 

Let   x  =  the   miles   per  hour   of    the   return;     then 

2_2  — 

=  a-{-x  =  3Q   hours   when  x  = 


a  —  x  a  —  x 

15,  and  the  number  of  hours  required  ;  and  it  appears 
that  the  messenger  has  farther  to  go,  and  goes  in  less  time, 
the  more  a  exceeds  x.  The  numerator  a  2  —  x  2  being  an 
expression  of  the  second  degree,  does  not  vary  uniformly 
with  a*,  hence  the  quotient  has  a  value  that  varies  not 
uniformly,  but  approaches  a  fixed  amount  for  x  =  a. 

3.  Required  the  value  of  —  when  x  —  0. 

X 

ax          a 

Ans.    -—  =  —  =  a. 

X  1 

170,  In  the  cases  which  have  thus  far  been  presented 
we  have  evidently  obtained  the  required  value  of  the  van 
ishing  fractions  by  reducing  the  fraction  to  its  lowest 
terms,  or  by  simple  algebraic  processes. 

4.  Required  the  value  of     *  ~^-_wben  x  =  a.      In 

«\/  X  "  — ~  Qt 

this  case,  if  we  divide  the  numerator  and  denominator  by 
— — —  and  the  resulting  quotients  by  Jx  —  a,  we  have 

the  following : 

x  —  a  x  —  a 


__  a2)  (J*  -f  Jo 


"_? — _  rr  0  when  x  =  a , 

.  (Jx  +  •/ a) 

because  we  have  removed  all  negative  quantities  from  the 


x  +  a 


VALUE   OP   VANISHING   FRACTIONS.  163 

denominator ;  to  produce  the  same  result  by  one  divisor, 
it  must  be 


a  divisor  that  is  very  far  from  being  quite  obvious.  So  that 
it  appears  that  these  algebraic  processes  cannot  have  in 
general  view  the  reduction  of  the  fraction  to  its  lowest  terms, 
but  to  effect  a  transformation  of  whatever  kind  that  may, 
as  above,  remove  ambiguity.  It  is  better,  therefore,  to 
adopt  a  direct  and  uniform  process  for  determining  the 
value  of  a  vanishing  fraction  :  this  process  the  calculus,  by 
differentiation,  supplies. 

171.  Since  all  algebraic  functions  of  a  variable  must 
vary  in  value  when  the  variable  does,  for  the  variable  is 
immediately  eliminable  from  all  expressions  containing 
it,  which  do  not  vary  when  the  variable  does,  such  as 

b  -\-  (a  —  a)  a3,  -  —  ,  etc.,  it  follows  that  the  sue- 

x  -\-  60 

cessive  differential  coefficients  of  a  function  cannot  all  be  0 
in  value,  or  vanish  for  a  particular  value  of  the  variable. 

In  the  case  of  fractions  vanishing  at  a  particular  value 
of  the  variable,  we  have  evidently  two  functions  of  one 
and  the  same  variable,  and  which  need  not  be  like  in  form, 
and  which  therefore  ought  to  be  designated  by  the  dis 
tinctions  of,  say,  F  x  for  the  numerator,  and  fx  for  the 
denominator  ;  this  gives  us,  in  view  of  the  hypothesis, 

Fx        ° 


Now,  in  case  F  x  and  fx  are  of  a  nature  to  be  developed 
by  Taylor's  Theorem  for  the  value  in  question,  let  them 
be  respectively  developed,  or  let  us  entertain  the  sugges 
tion  of  each  of  their  values  moving  out,  as  it  were,  from. 
zero  by  the  nearest  appreciable  amount,  as  when  x  should 


164  DIFFERENTIAL   CALCULUS. 

be  x  -f-  h ;  then  we  have,  instead  of  zero  for  F  x,  and  for 
/"ic,  certain  indefinitely  small  compared  quantities,  if  one 
or  the  other  should  not  still  remain  zero ;  in  which  case, 
we  have  what  we  seek  for,  in  0  or  co  ,  as  the  value  of  the 
fraction.  Understanding,  now,  that  F  x  is  y  in  Taylor's 
Theorem,  and  that/ x  is  some  other  and  a  different  y,  and 
agreeing,  for  convenience,  to  represent 

*JL  byy,  Jl*.  by i/',  -    d3y       by //",  etc. 

dx     J  r  '  dx2A.2     J2     'd*3.1.2.3     J  l 

dy  d2  y  d3  y 

where  the  number  of  accents  is  made  to  agree  with  the 
order  of  the  dif.  coefs.  in  numerical  name,  we  have 

Fx+p'  h+p"  h*+p'"  A3  +  etc. 
fx  +  q'  h  +  q"  h*  +  q'"  h3  +  etc.' 

Now,  since  by  hypothesis  l<Jx  =  Q,fx  =  Q,  at  the 
value  in  question,  they  severally  become  of  no  account  in 
the  fraction  of  the  development,  and  may  be  expunged,  so 
that  we  have,  after  dividing  by  A, 

F  (x  +  A)  p'  +p"A+p";A2  +  etc.  ^ 

f  (X  +  A)         ?'  +  0"  A  +  j'"  AZ  -(.  etc. J 

and  when  A  =  0, 

Fx       P' 


as  the  required  value  of  the  vanishing  fraction ;  but  pos 
sibly  —  may  become  — ,  in  which  case  we  may  expunge 

-  from  equation  (2.),  which  then  becomes,  after  divid- 
tf 

ing  by  A, 

"       '" 


VALUE   OF   VANISHING   FRACTIONS.  165 


which  becomes,  when  h  =  0, 

Fx        p" 


as  the  required  value  in  case  it  does  not  become  -  ;  in 
which  case  expunge  —  from  (3.),  and  divide  by  A,  and  we 

have,  when  h  =  0, 

Fx  _  p'" 
Jx  ~  "p 

as  the  required  value,  if  it  is  any  thing  else  than  -  ;  and  so 
on,  so  that  we  have  the  following  rule  for  determining  the 
value  of  a  fraction  of  which  the  numerator  and  denomina 
tor  vanish  when  x  takes  a  particular  value  : 

172.  For  the  numerator  and  denominator  substitute 
their  first  dif.  coefs.,  their  second  dif.  coefs.,  and  so  on, 
till  we  obtain  the  first  fraction  of  which  both  numerator 
and  denominator  do  not  vanish,  for  the  required  value  of 
x  ;  this  fraction  is  the  value  required. 

We  have  already  shown  that  we  must  arrive  at  such  a 
fraction. 

5.  Required  the  value  of  —  -  —  -  —  when  x  =  a. 

(3  a  —  3  a;)3 

pm         _  24  (a  -  x)  0 

Ans.   —  —  -  =  —  =  0  . 
qin        —       162  162 

6.  Required  the  value  of  -  when  x  =  1. 

4  x  3  —  12  x  +  8 

Ans.  co  ,  by  1st  dif.  coefs. 

X3  _  a3 

7.  Required  the  value  of  -  when  x  =  a. 

Xz  —  «2 

Ans.   3  a,  by  2d  dif.  coefs. 


166  DIFFERENTIAL   CALCULUS. 


8.  Required  the  value  of  -  —-when  x  =  1. 

(1  —  *)3 

Ans.   ---  by  2d  dif.  coefs. 

3 

9.  Required  the  value  of     4j^6a.a  —  ^TT3  wben  x  ==  1- 

Ans.   </>  by  2d  dif.  coefs. 

173.  Inasmuch  as  we  have   deduced  the   process   for 

Fx  0 

finding  the  value  of  —  when  it  becomes  -,  we  have,  in 
fx  o 

effect,  found  the  process  for  finding  the  value  of  -  -  X  — 

(of  which  expression  the  first  factor  is  certainly  0,  and  the 
second  is  co),  that  is,  for  finding  the  value  of  a  product  of 
two  functions  as  factors,  one  of  which  becomes  0  and  the 
other  co  when  the  variable  takes  a  particular  value.  We 
have  only  to  take  the  first  as  a  numerator  and  the  recip 
rocal  of  the  second  as  denominator  and  we  have  the  van 
ishing  fraction  just  investigated. 

10.  Required  the  value  of  (xn  —  1)  X  --  when  x  =  I. 

Ans.   n. 

174.  The  foregoing  demonstration  of  the  process  for 
finding  the  value   of  a  vanishing  fraction  embraces  the 
principle  of  finding  the  value  of  a  fraction  which  becomes 
^  under  the  same  condition  ;  for  any  fraction  is  the  same 
in  value  as  the  reciprocal  of  its  denominator  taken  for 
numerator,  and  the  reciprocal  of  its  numerator  taken  for 
the  denominator.     Thus,  if 


then       = 

fx  x  _ 

Fxfx  Fx 


VALUE   OP   VANISHING    FRACTIONS.  167 

We  remark  that  evidently  the  reciprocal  of  an  infinite 
quantity  is  zero,  i.  e.,  —  =  0. 

11.   Required  the  value  of  ---  '-  —  -    -  when  x  =  a. 

a  —  x         aa  —  x'" 

Ans.   5  a. 

175.  Lastly,  the  demonstration  ateo  embraces  the  case 
of  determining  the  value  of  the  difference  of  two  functions, 
each  of  which,  for  a  particular  value  of  the  variable,  be 
comes  infinite  ;  for,  in  subtraction,  any  remainder  is  equal 
the  fraction  of  which  the  numerator  is  the  excess  of  the 
reciprocal  of  the  subtrahend  above  the  reciprocal  of  the 
minuend,  and  the  denominator  is  the  reciprocal  of  the 
product  of  minuend  and  subtrahend.  Thus,  Fx  being  co, 
audfx  being  co, 

Fx—fx  1 


11.   Required  the  value  of  ---  when  x  —  a. 

x*  —  a3         x  —  a 

Ans.   to. 

176,  Whenever  an  infinite  quantity  is  generated  by  the 
denominator  of  a  fraction  becoming  0,  since  —  0  =  -f-  0, 
it  is  evident  that  such  infinite  quantity  has  the  ambiguous 
sign  ±  ,  and  becomes  ±  co  . 

177.  The  same  characteristics  of  different  values  belong 
to  infinite  quantities  that  belong  to  finite  quantities  and  to 
zero,  dependent  upon  their  mode  of  generation,  or  of  rela 
tion  to  each  other  by  factors,  by  radical  expressions,  or 
otherwise.     It  is  not  considered  that  a  finite  quantity  is 
any  addition  to  an  infinite  one,  or  diminution  from  one, 
and  such  finite  quantity  may  be  expunged. 


168  DIFFERENTIAL   CALCULUS. 

178.  An  infinite  quantity  is  an  impossible  quantity  ; 
hence  all  conclusions  predicated  on  the  possibility  of  an 
infinite  quantity  must  fail ;  with  the  exception,  however, 
that  certain  conclusions  are  practicable  with  reference  to 
the  finite  terms  of  a  series.  But  general  developments 
fail  for  certain  values  of  a  quantity  when  any  term  of 
a  series  becomes  infinite  for  such  values. 


SECTION  XXII. 

EXCEPTIONAL    PRINCIPLE    RELATING    TO    TAYLOR'S 
THEOREM. 

179,  The  general  development  of  every  function  of  a 
variable  according  to  the  ascending  entire  and  positive 
powers  of  its  increment,  for  general  values  of  the  variable, 
is  possible  by  Taylor's  Theorem.  But  this  development 
does  not  hold  whenever,  for  a  particular  value  of  the  vari 
able,  any  of  the  coefficients  of  Taylor's  series  become  infi 
nite.  Thus,  the  general  development  of  Fx-=.  \j '  (x  —  a), 
when  x  is  replaced  by  x  -\-  A,  is,  by  this  theorem, 

F  (x  +  h  —  a)  = 
(x  +  a)*  +  i  (x  -  a)~*  h  —  -  (x  —  a)'1  h*  +  etc., 

/  o 

where,  in  case  a?  —  a,  all  the  differential  coefficients  become 
infinite.  And  it  will  be  observed  that  when  any  dif.  coef. 
becomes  infinite,  all  succeeding  ones  do  also. 

It  is  not  held  that  the  development  fails  when  these 
coefficients  become  imaginary  if  the  variable  takes  a  par 
ticular  value ;  because  the  function  of  x  -\-  h  would  itself 
become  imaginary  at  the  same  value,  and  it  is  proper  that 
one  imaginary  quantity  should  be  equated  with  another. 


FAILURE   OF  TAYLOR'S   THEOREM.  169 

Since  such  coefficients  as  become  infinite  for  the  pro 
posed  value  of  the  variable,  become  so  on  the  principle  of 
the  denominator  of  a  fraction  vanishing,  the  sign  of  such 
infinite  coefficient  becomes  always  ±  ,  or  is  ambiguous. 

Whatever  uses,  therefore,  we  may  on  general  principles 
wish  to  make  of  Taylor's  Theorem,  become  exceptional  on 
the  condition  alluded  to.  Thus,  in  regard  to  maxima  and 
minima,  suppose  it  were 

1.  Required  to  find  the  maxima  or  minima  values  of  the 
function  y  =  b  -f-  (x  —  «)l. 

...££=!(«._«)», 

dx         3   v 
4 


dy 

The  equation  —  =  0  gives  x  =  a  ;  and  if  we  proceed 

d  x 

to  determine  whether  we  have  a  maximum  or  minimum  for 
x  =  a,  we  find  that  for  this  value  the  second  dif.  coef.  is 
infinite,  which  is  but  another  name  for  an  impossible  quan 
tity.  But  as  we  have  never  any  thing  to  do  with  the 
greatness  of  a  dif.  coef,  when  we  examine  it  with  the  pur 
pose  we  now  have  in  view,  but  have  to  do  with  its  sign 
only,  this  second  dif.  coef.  must,  from  its  mode  of  deriving 
its  infinite  value,  have  the  ambiguous  sign  ±  ;  therefore  the 
function  in  question  must,  at  x  =  a,  be  inferred  to  have 
both  a  maximum  and  a  minimum,  which  nevertheless  is 
still  a  possibility  for  some  functions,  but,  with  reference  to 
the  one  in  question,  may  be  found  by  algebraic  or  arith 
metical  tests  not  to  be  true,  but  that  there  is  only  a  mini 
mum.  For  if  we  test  the  value  of  y  immediately  before 
x  =  a,  as  when  x  =  a  —  A,  and  immediately  after  x  =  a, 
as  when  x  =  a  -|-  A,  that  is,  by  substituting  a  ±  h  for  x 
in  the  function,  we  shall  have 

F  (a  ±  h)  =  5  +  At. 
15 


170  DIFFERENTIAL   CALCULUS. 

Hence  b  is  increased  for  either  sign  of  h  ;  consequently 
x  ==.  a  renders  the  function  a  minimum  when  having  the 
value  b. 

It  will  be  observed  that  since  the  odd  root  of  a  negative 
quantity  is  possible,  such  root  being  negative,  and  the  even 
power  of  all  quantities  is  positive,  the  fourth  power  of  the 
third  root  is  positive. 

180.  To  obtain  the  true  development  of  a  function  for 
that  one,  or  those,  particular  values  of  the  variable  which 
cause  Taylor's  Theorem  to  fail,  the  usual  course  is  to  recur 
to  the  ordinary  process  of  common  algebra,  after  having 
substituted  a  -\-  h  for  x  in  F  x. 

2.  Required  the  development  of  F  x  =  2  ax  —  a?2  -|- 

«  V^2  —  «2  for  the  condition  when  a?  becomes  a  -f-  h  ;  and 
to  arrange  the  terms  according  to  the  increasing  exponents 
of  A. 

Substituting  a  -f-  h  for  x  we  have 

F  (a  +  A)  =  a*  —  h*  +  ah*  (2  a 


developing  the  binomial  (2  a  +  A)  *  by  the  Binomial  The 
orem,  and  multiplying  its  terms  by  a  AS,  we  have 


8(2a) 

+  etc. 

The  algebraic  process  in  question  is  any  that  will  reduce 
complex  terms  to  simple  ones,  in  which  h  shall  appear  as 
a  factor  writh  any  whole  or  fractional  index,  unless,  per 
haps,  it  may  as  above  be  eliminated  from  any  term  or 
terms.  It  will  be  observed  that  these  various  coefficients 
of  A  are  not  differential  coefficients. 

3.  Required  to  determine  whether  y  in  the  following 
F  (cc,  y)  •=.  0  has  a  maximum  or  minimum,  viz.  : 


FAILURE   OF  TAYLOR'S   THEOREM.  171 


d_y__ 

d  x  ~     3      (y  —  6)  *          3   (x  —  a] 

In  this  instance  we  might,  on  inspecting  its  first  form, 

dy 
incautiously  infer  that  —  becomes  0  when  x  —  «,  if  we 

d  x 

regard  only  the  numerator  (Art.  112).  But  we  are  obliged 
to  inquire  whether  we  have  not  here  a  vanishing  fraction, 
the  denominator  becoming  0  when  x  =.  a,  which  we  should 
find  to  be  true,  and  that  its  value  is  infinite  when  x  =  a ; 
which  we  see  at  once  on  inspecting  the  second  form.  And 
we  find  that  x  is  infinite,  by  the  rule  of  vanishing  fractions, 
when  the  first  dif.  coef.  =  0,  and  therefore  (Art.  108)  we 
should  have  no  maximum  or  minimum.  If,  nevertheless, 
we  consider  the  infinite  value  of  the  first  dif.  coef.,  we  find 
it  occurs  when  x  =  a,  and  at  this  value  Taylor's  Theorem 
fails.  Yet  if  we  test  this  value,  x  —  a,  in  the  function,  or 
rather  the  values  of  a;,  within  h  of  a,  we  shall  find  a  mini 
mum.  Substituting  a  ±  h  for  x  we  have  for  y 


and  b  is  increased  for  either  sign  of  A,  since  all  possible 
values  ofy  are  b  and  something  additional  to  #;  hence  a 
minimum.  Hence  an  important  principle  supplementary 
to  our  Section  on  Maxima  and  Minima,  which  is  : 

181.   Before  we  can  conclude  in  any  case  that  the  val- 

d  y 

ues  of  x  deduced  from  the  condition  —  =0  comprise 
among  them  all  those  that  can  render  a  function  a  maximum 
or  minimum,  we  must  examine  those  values  of  x  arising  from 

the  condition  —  —  c/> .    And  as  this  is   a  case  wThere  the 
d  x 

development  by  Taylor's  Theorem  fails,  we  must  make  this 


172  DIFFERENTIAL   CALCULUS. 

examination  by  the  algebraic  method  of  substituting  each 
of  these  values,  as  affected  by  the  addition  and  subtraction 
of  A,  for  x,  in  the  proposed  function,  and  observing  which 
of  the  results  agree  with  the  conditions  of  maxima  and 
minima,  as  by  definition. 

182.  The  method  pointed  out  in  a  previous  section  for 
determining  the  value  of  a  vanishing  fraction  now  requires 
the  mention,  that  in  case  a  numerator  or  denominator  fails 
to  be  developable  by  Taylor's  Theorem,  we  must  adopt  the 
algebraic  method  of  development,  for  either  or  both  which 
so  fail.  We  should  then  arrange  the  terms  as  numerator 
and  denominator  according  to  the  increasing  exponents  of 
h  ;  then  divide  each  term  in  '  either  by  h  at  the  lowest 
power  of  either  ;  then  test  what  the  fraction  becomes  for 
h  =  0.  Whatever  value  it  has  is  the  value  desired. 

4.  Required  the  value  of  -  —  when  x  =  a. 


(2  a  A){j  +-    (2  a  A)4A«  +  ,  etc. 


hi  hi 

2=(2«)»,     An, 


5.   Required  —  in  y  =  x  +  (x  —  a)  2  V  'x  for  x  —  a. 

d  x 


6.  Required  -  -  in  y  =  x  -\-  (x  —  a)  2  V#  for  x  =  a. 

Ans.   ±  2  Va^ 

7.  Required  —  and  -  -  in  (y  —  x)  2  —  (x  —  a)  4  x  for 

d  x  d  #2 


NATURE   OF   LOGARITHMS.  173 

It  is  worthy  of  notice  whether  the  last  implicit  function 
is  not  virtually  the  preceding  explicit  one  when  we  re 
gard  y. 

8.   Required  —  in  y 3  =  (x  —  a) 3  (x  —  b)  for  x  —  a. 

d  x 

Ans.    (a  —  b)  3 . 

We  here  terminate  the  portion  of  our  treatise  relating 
strictly  to  Algebraic  Functions. 


SECTION  XXIII. 

NATURE   OF  LOGARITHMS  AND  EXPONENTIAL   QUAN 
TITIES. 

183,  Thus  far  we  have  made  no  mention  of  any  func 
tions  in  connection  with  which  the  variable  occurs  as  an 
exponent,  whether  of  a  power,  as  in  ax,  or  of  a  root,  as  in 

5^,  or  of  a  power  and  a  root  which  is  the  characteristic  of 
a  fraction  in  general  as  exponent.  The  reason  of  this  has 
been  a  regard  to  a  distinctive  division  of  subjects.  The 
discussion  of  the  properties  and  differentiation  of  strictly 
algebraic  functions  being  completed,  we  shall  be  brought 
to  consider,  in  the  succeeding  section,  functions  of  a  new 
order.  But  we  are  not  to  forget  that  the  subjects  strictly 
relate  to  numerical  analysis,  and  are  directly  consecutive 
with  our  preceding  inquiries.  It  is,  however,  called  trans 
cendental  analysis,  as  indicative  of  being  of  a  grade  above 
what  is  commonly  called  algebraic. 

Since  many  treatises  of  algebra,  otherwise  quite  com 
plete,  do  not  contain  an  account  of  the  theory  and  uses  of 
15* 


174  DIFFERENTIAL  CALCULUS. 

logarithms,  and  it  would  be  unfortunate  for  us  to  go  for 
ward  without  such  preparation,  we  will  devote  this  section 
to  the  Nature  of  Logarithms  and  Exponential  Quantities. 

184.  A  logarithm  is  such  exponent  as,  applied  to  any 
number  a  greater  than  1,  shall  cause  a  number  to  be  de 
noted  equal  to  any  positive  numerical  quantity  b  what 
ever,  greater  or  less  than  1.     The  logarithm  in  question 
is  the  logarithm  of  #,  the  latter  number. 

185.  A  system  of  logarithms  is  a  collection  of  exponents 
such  as  offer  by  selection,  one  which,  when  applied   to 
some  constant  number  (originally  arbitrary),  called   the 
base,  will  render  this  base  the  equal  and  the  representative 
of  any  number   whatsoever;    and  hence    every   number 
whatsoever.     This  exponent  is  the  logarithm  of  the  latter 
number. 

186.  In  accommodation  to  the  decimal  system  of  num 
bers,  the  number  10  has  been  selected  as  the  base  of  the 
common  system.     Accordingly,  in  this  system  1  is  the  loga 
rithm  of  10,  because  10  *  =  10 ;  2  is  the  logarithm  of  100, 
because  10 2  =  100;  3  of  1000,  because  10 3  =  1000,  etc. 
Zero  or  0  is  the  logarithm  of  1,  because  10°  —  1.     As  it  is 
seemingly  arbitrary  to  call  0  the  common  logarithm  of  1, 
we  remark  that  it  is  strictly  inferred  from  the  ratio  by 
which  logarithms  diminish  with  entire  units.     We  have 
for  logarithms,  by  continuity,  the  following,  placed  in  con 
nection  with  the  numbers  of  which  they  are  the  loga 
rithms  : 

3  2             1           0—1-2          -3,     etc. 

1000  100          10          1         df         yfo         TTyW>     etc. 

or,  ,1          ,01          ,001  ,     etc. 

whence  it   appears  that  while  all  the  natural  numbers. 


NATURE   OF  LOGARITHMS.  175 

when  selected  for  their  logarithms  in  entire  numbers,  vary 
successively  by  the  ratio  10  or  T\y,  their  logarithms  vary 
by  a  uniform  difference  of  1.  And  it  appears  that  we  pass 
zero  with  the  preservation  of  this  law. 

It  already  appears  that  the  logarithm  of  a  fraction,  by 
which  we  mean  a  numerical  quantity  less  than  1,  is  nega 
tive.  If  we  represent  the  logarithm  of  0  by  —  n,  we 
have 

io-  =  o  =  i., 

which  requires  that  the  denominator  10 w  be  infinite,  or 
what  is  the  same,  n  to  be  infinite. 

It  further  appears  that  the  common  logarithm  of  any 
number  greater  than  1  and  less  than  10  must  be  between 
0  and  1,  i.  e.,  be  a  fraction,  or  rather  we  should  find  it  not 
to  be  expressible  exactly  even  as  a  fraction.  The  same 
remark  applies  to  any  logarithm  which  is  not  a  whole  num 
ber,  positive  or  negative ;  that  is,  the  logarithm  of  any 
number  intervening  between  1000  and  100 ;  100  and  10 ; 
10  and  1  ;  1  and  ,1 ;  ,1  and  ,01 ;  ,01  and  ,001 ;  etc.,  is  ex 
pressed  by  a  whole  number,  or  0,  and  a  fraction,  most  con 
veniently  a  decimal  fraction.  Such  decimal  fraction  not 
absolutely  expressing  the  value  intended,  is  what  is  known 
as  an  irrational  quantity.  Thus,  the  common  logarithm  of 
2  is  0,3010299,  that  of  543  is  2,7352793,  etc.  In  view  of 
this  mcommensurableness  of  most  numbers  and  their  re 
spective  logarithms,  only  an  approximate  definition  can  be 
given  of  a  logarithm  in  general.  The  definition  should 
embrace  a  reference  to  a  certain  power  of  a  certain  root  of 
the  understood  base,  by  successive  degrees  of  proposed 
approximation,  each  replacing  the  preceding.  Thus  the 
common  logarithm  of  2  is,  by  the  degree  of  nearness  de 
sired, 

3  301  30102  301029 

10'     or     woo'     or     io^o'     or     1000000'     etc-; 


176  DIFFERENTIAL   CALCULUS. 

which  are  to  be  read,  10th  root  of  the  3d  power  of  the 
base,  or  the  1000th  root  of  the  301st  power,  etc.  In  this 
way  must  we  enunciate  the  significance  of  a  decimal  frac 
tion  as  an  exponent.  In  proper  mathematical  expression, 
as  we  shall  see,  logarithms,  when  not  entire  numbers,  are 
discoverable  as  existing  in  an  infinite  series,  which  indeed 
a  decimal  fraction  not  terminating,  itself  is. 

The  method  of  using  logarithms  for  the  purpose  of  fa 
cilitating  operations  with  numbers  is  quite  evident  after 
the  study  of  algebraic  exponential  quantities. 

187.  In  order  to  multiply  quantities,  we  add  their  loga 
rithms  ;  the  sum  of  their  logarithms  is  the  logarithm  of  the 
product,  or  continued  product,  of  two  or  more  quantities. 

Hence  any  number  is  multiplied  by  10  by  adding  1,  the 
common  logarithm  of  10,  to  that  of  the  number;  this 
logarithm,  thus  increased,  becomes  the  logarithm  of  the 
product.  It  is  multiplied  by  100  by  adding  2,  by  1000 
by  adding  3  to  its  logarithm.  Advantage  is  taken  of  this 
property  in  the  preparation  of  tables  to  insert  only  the 
fractional  part  of  a  logarithm,  leaving  the  integral  part,  or 
characteristic,  to  be  extemporized  according  to  (being  one 
less  than)  the  number  of  places  of  the  integral  pan  of  the 
number  of  which  the  logarithm  is  desired.  Thus,  the 
logarithm  of 


54360 

is 

4,7352794 

5436 

is 

3,7352794 

543,6 

is 

2,7352794 

54,36 

is 

1,7352794 

5,436 

is 

0,7352794 

,5436    is        1,7352794 

The  logarithm  of  ,5436  is  expressed  here  with  its  decimal 
part  positive,  while  its  integral  part  is  negative.  The 
practice  is  in  common  use  of  adding  10  to  the  characteris- 


NATURE  OF  LOGARITHMS.  177 

tic  in  such  cases,  since  the  inconvenience  of  the  negative 
characteristic  is  thus  avoided,  and  no  error  would  be  likely 
to  arise  in  common  uses  which  would  not  be  strikingly 
obvious,  and  easily  corrected  by  subtracting  the  10.  Thus 
the  logarithm  of  ,5436  is  expressed  as  9,7352794. 

188.  Division  of  numbers,  being  the  converse  of  multi 
plication,  is  effected  by  the  subtraction  of  the  logarithm  of 
one  number  from  that  of  the  other ;  this  difference,  when 
positive,  is  the  logarithm  of  the  number  of  times  the  less  is 
contained  in  a  greater  number ;  when  negative,  it  is  the 
logarithm  of  the  fractional  time  the  greater  is  contained  in 
the  less.     We  have  already  found  that  the  logarithm  of 
a  fraction  is  negative. 

189.  In   order  to  raise   a  numerical   quantity  to  any 
power,  we  multiply  the  logarithm  of  that  quantity  by  the 
number  denoting  the  power  required ;  the  product  is  the 
logarithm  of  the  power  required. 

190.  In  order  to  extract  any  root  of  a  numerical  quan 
tity,  we  divide  its  logarithm  by  the  cardinal  number  ex 
pressing  the  root  required  in  the  ordinal  form  of  expres 
sion,  as  2  for  second,  etc. 

The  method  t  of  finding,  by  the  use  of  logarithms,  the 
fourth  term  of  a  set  of  common  direct  proportionals,  is 
therefore  extremely  obvious ;  we  add  the  logarithms  of  the 
first  and  second  terms ;  from  the  sum  subtract  that  of  the 
third ;  the  remainder  is  the  logarithm  of  the  fourth,  or 
term  required. 

A  small  volume  is  to  be  obtained  containing  a  table  of 
common  logarithms,  for  all  numbers  from  0  to  10000,  to  six 
places  of  decimals,  sometimes  to  seven  places.  The  meth 
od  of  taking  out  the  logarithm  for  any  number  within  these 
limits,  and  of  extending  the  use  of  the  table  to  much 
greater  numbers,  as  well  as  of  finding  the  natural  number 


178  DIFFERENTIAL   CALCULUS. 

corresponding  to  any  possessed  logarithm  is  usually 
printed  with  the  table.  The  table  is  also  to  be  found 
in  treatises  of  navigation  and  surveying. 

The  logarithm  of  a  negative  quantity  does  not  belong 
to  the  same  system  with  those  of  positive  quantities. 
When,  however,  certain  numerical  operations  with  nega 
tive  quantities  are  to  be  done,  we  may  eliminate  the  con 
dition  of  their  negativeness  until  the  result  is  reached, 
when  the  appropriate  sign  may  be  prefixed  to  it,  as  alge 
braically  determined. 


SECTION  XXIV. 

DIFFERENTIATION    AND    DEVELOPMENT    OF    LOGA 
RITHMIC    AND    EXPONENTIAL    FUNCTIONS. 

191.  A  logarithmic  function  is  the  logarithm  of  a  vari 
able  quantity  ;  as,  log.  ic,  or  log.  (b  -\-  x  n),  which  do  not  de 
note  the  logarithm  which  is  £c,  etc.,  but  the  logarithm  of  the 
number  which  is  #,  etc. 

192,  An  exponential  function  is  one  in  which  the  vari 
able,  or  some  function  of  it,  holds  the  position  of  index  or 
exponent;  as,  «x,  or  bnx,  the  root  being  a  constant. 

It  is  important  to  observe  that  such  index,  when  con 
sidered  as  some  logarithm,  is  not  the  logarithm  of  the 
same  quantity  to  which  it  is  attached  as  index,  but  of  the 
entire  power  which  itself  is  employed  in  expressing. 

A  number  or  numerical  quantity,  and  its  logarithm,  are 
distinguished  as  natural  number,  or  quantity,  and  its  log 
arithm. 


LOGARITHMIC   FUNCTIONS.  179 

193,   We  now  proceed  to  differentiate 
y  =  log.  x, 

for  any  system  of  logarithms  having  for  its  base  a,  which 
convenience  will  require  to  be  considered  greater  than  1  ; 
for  we  immediately  derive,  by  the  converse  of  the  defini 
tion  of  a  logarithm, 

«=«-,  (i.) 

and  by  no  variation  of  y  while  positive  can  av  represent  all 
numerical  quantities,  unless  a  be  greater  than  1. 

Letting  y  take  the  increment  h  as  the  independent 
variable,  and  x'  denote  the  corresponding  value  of  #,  we 
have 


Let  us  now  substitute  "L-\-b  for  a,  and  develop 
by  the  binominal  theorem,  and  we  have 

.—  V 


h    h-l   A-2_3    . 

7'—  •  —  *  +'eto- 

The  multiplication  of  A  (A  —  1)  (A  —  2),  etc.,  being  done, 
and  all  the  quantities  selected  from  the  successive  terms 
of  the  continued  series,  which  are  factor  to  A,  and  placed 
or  indicated  within  the  following  parenthesis,  and  sA2 
being  used  for  all  succeeding  terms,  in  which  s  alone 
includes,  in  some  sense,  A,  or  a  series  involving  A,  we 
have 

s  A2; 


180  DIFFERENTIAL  CALCULUS. 

multiplying  both  members  by  a»,  and  calling  the  terms 
within  the  parenthesis  c,  we  have 


Subtracting  the  equals  x  =  a",  from  the  above  equals, 
we  have 


-\-sav  h~ 

x'  —  x 


which  becomes,  when  A  =  0, 

(2.) 


dy 
Now,  ay  being  =  x  by  (1),  we  have 

dx 
^==C£C; 

dx        c       x  ' 

and    dy  =  —  X  — 

c          a; 

where  b  being  =  a  —  1,  we  have 
1  1 


ca-l-^(a-l)2  +  J(a-l)3-i(a-l)4+e1 

Defining  now—,  or  the  reciprocal  of  c,  as  the  modulus 
of  the  system  of  logarithms  which  has  the  number  a  for 
its  base,  we  have  the  general  rule : 

194.  To  differentiate  a  logarithmic  function,  or  the 
logarithm  of  a  variable  quantity,  we  must  multiply  the 
modulus  of  the  system  by  the  differential  of  the  natural 
quantity ',  and  divide,  the  product  by  the  natural  quantity 
itself. 


EXPONENTIAL  FUNCTIONS.  181 

195.  Since  a  logarithmic  function  may  be  of  a  more 
complex  form  than  simply  log.  jc,  some  function  of  a;  being 
in  the  place  of  #,  we  must  evidently  have  in  the  result,  in 
the  place  of  dx,  the  whole  differential  of  such  function,  in 
which,  indeed,  dx  will  be  found  as  a  factor. 

196.  If  the  variable  quantity  of  which  the  logarithm  is 
intended  in  the   constitution  of  a   logarithmic   function, 
contain  a  constant  factor,  since  such  factor  will  be  found, 
in  pursuance  of  the  rule  for  differentiation,  as  still  a  factor 
of  the   numerator    and  of  the  denominator,  it  becomes 
evident  that  such  factor  contributes  nothing  affecting  the 
differential.     This  is  consistent  with  a  previous  change  of 
the  function  into  the  sum  of  the  logarithms  of  the  factors  ; 
thus  : 

d  [log.  b  (a  —  x)  ]  =  d  [log.  b  +  log.  (a  —  x)  ]  = 


197.  But  if  the  logarithmic  function  be  associated  with 
a  factor  in  a  manner  oy  which   the  logarithm  of  such 
product  is  not  intended,  the  constant  factor  will  be  found 
affecting  the  differential  as  factor  ;  thus, 

d  [m  log.  x]  —  m  d  log.  x  =  --  .— 

X          C 

198.  Selecting  from  the  demonstration  of  the  differen 
tiation  of  logarithmic  functions,  equation  (1),  viz., 


we  observe  a  v  to  be  an  exponential  function,  and  that  its 
differential  coefficient  is  expressed  in  equation  (2),  y  being 
the  independent  variable,  and  x  being  dependent.  It  is 

dx 

—  =  cav 

dy 

/.  dx=:cav  dy. 
16 


182  DIFFERENTIAL   CALCULUS. 

199,  Therefore,  to  differentiate  an  exponential  function, 
we  must  multiply  together  the  reciprocal  of  the  modulus 
of  the  system  of  logarithms,  determined  by  the  base  of  the 
exponential,  the  exponential  itself,  and  the  differential  of 
the  variable  exponent. 

200.  An  exponential  function  is  not  considered  to  be 
restricted  to  the  very  simple  form  of  ay,  or  indeed  to  be 

restricted  at  all,  for  bay  ,   ay  +  bna  ,  -  v  -I-  bv,  etc.,  are 

(avc) 

held  to  be  examples  ;  the  base  must  be,  however,  the  root 
of  the  power  indicated,  and  in  b  ay  ,  b  is  no  part  of  the 
base,  nor  even  a  factor  of  it,  but  is  a  factor  of  the  power 
only. 

If  in  the  preceding  demonstration  we  had  at  the  outset, 


we  should  find  the  factor,  b,  passing  through  the  demon 
stration,  and  appearing  in  the  result, 

dx  =  c  b  av  dy; 

in  the  sequel  we  shall  have  a  practical  use  for  this  obser 
vation. 

201.  For  the  hyperbolic  or  Napierian  system  of  log 
arithms,  the  modulus  has  been  assumed  —  1,  which  value 
renders  its  reciprocal  =  1.     As  to  this  system,  therefore, 
the  mention  of  the  modulus  may  be  eliminated  from  the 
two  preceding  italicized  rules  ;  if  in  the  succeeding  context 
all  mention  of  a  modulus  is  omitted  in  any  operation,  the 
hyperbolic  system  will  be  understood  to  be  intended. 

202.  For  c  —  1,  the  value  of  a,  as  the  base  of  the  hyper 
bolic  system,  must  be  deduced.    We  will  therefore  de- 


EXPONENTIAL   FUNCTIONS.  183 

velop,  by  Maclaurin's   Theorem,  the  exponential  function 
a  v  according  to  the  powers  of  y. 

Let    x  =  a  v  .-.  when  y  =  0,  x  —  1 ; 

dx 

-  =  c  a  y  .:  when  y  •=.  0,  x  =  c : 

dy 


dy3 

etc.  etc. 


When,  therefore,  c  =  1  and  y  —  1,  we  have 

a^l  +  l-fl-l-—1-^  --  .  --  1_,  etc. 
r  2    '     2  .3    '2.3.4^ 

=:  2^71828, 

which  is  the  base  of  the  hyperbolic  system. 

But  if  we  wish  to  assume  a  —  10,  which  is  very  desira 
ble  for,  and  is  the  base  of  the  common  system,  since  a  — 
1  =z  9,  we  have  for  c 


=  2,  30258509  ; 

and  -  —  ,43429448,  which  is  the  modulus  of  the  common 
c 

system. 

20.3.   For  the  common  system  of  logarithms,  therefore, 
the  fraction  ,43429448,  must  be  read  as  modulus  in  the 


184  DIFFERENTIAL  CALCULUS. 

foregoing  rules  for  the  differentiation  of  logarithmic  func 
tions,  (Art.  194) ;  and  the  number,  2,30258509  as  the  recip 
rocal  of  the  modulus  in  the  rule  for  the  differentiation  of 
exponential  functions  (Ait.  199.). 

To  recapitulate  :  we  have  for 

(  Modulus  of  hyp.  system,  assumed,      .     .     .  1, 

}  Base  of  the  hyp.  system  deduced,      .     .     .2,71828 

(  Reciprocal  of  this  modulus, 1, 

SBase  of  the  com.  system  assumed,     .     .     10, 
Modulus  of  com.  system  deduced,      .     .     .      ,43429448 
(  Reciprocal  of  this  modulus, 2,30258509 

Since  from  equation  (3)  foregoing,  which  is 


we  derive 

dy 1 

~dx        7 

which  becomes,  when  x  =  1, 


dx~  c> 

we  have  for  the  dif.  coef.  of  the  logarithm  of  1  in  every 
system,  the  modulus  of  such  system.  The  modulus  of  a 
system  is  therefore  the  ratio,  or  rate,  at  which  positive 
logarithms  come  into  being.  And  this  ratio  is  constant 
for  whatever  number  x  may  be.  Accordingly,  whatever 
the  number  may  be,  the  ratio  of  its  logarithms,  by  different 
systems,  is  always  constant.  Hence,  we  may  find  the 
hyperbolic  logarithm  of  any  number  from  its  common  log 
arithm,  by  multiplying  the  latter  by  2,30258509.  And, 
conversely,  a  hyperbolic  logarithm  may  be  converted  into 
a  common  logarithm  by  dividing  it  by  2,30258509. 


LOGARITHMIC   FUNCTIONS.  185 

In  regard  to  negative  logarithms,  or  those  of  fractions, 
or  any  numerical  quantities  less  than  1,  the  pursuance  of 
this  multiplication  renders  the  hyperbolic  the  less  loga 
rithm  than  the  common,  the  greater  negative  being,  of 
course,  the  less  quantity. 

For  the  most  concise  method  of  indicating  whether  the 
hyperbolic  or  common  logarithm  is  intended,  authors 
agree  to  cite  the  hyperbolic,  by  the  small  Roman  letter  1., 
or  log.,  the  common,  by  the  Roman  capital  L.,  or  Log. 
We  shall  adopt  this  distinction  hereafter,  when  distinction 
is  necessary. 

Mr.  J.  R.  Young  suggests  that  (log.)  2  x  shall  be  taken 
to  signify  log.  log.  a?,  or  logarithm  of  the  logarithm  of  cc, 
but  log.2  #,  having  no  parenthesis,  to  signify  the  second 
power  of  log.  x.  Sufficiently  explicit  is  log.  cc2  for  the 
logarithm  of  x2.  We  will  adopt  this  use. 

When  —  is  a  quantity  less  than  one,  log.  —  is  negative 
without  the  expression  by  a  negative  sign ;  hence  f —  log.  —  J 

becomes  a  positive  quantity ;  hence  log.  (  —  log.  —  j  is  the 
logarithm  of  a  positive  quantity,  and  becomes  entitled  to 
the  abridgment,  — (log.)2—.  The  succeeding  context 

presents  a  case  of  this  use. 
16* 


186  DIFFERENTIAL   CALCULUS. 


SECTION  XXV. 

EXAMPLES  OF   DIFFERENTIATION  AND   DEVELOPMENT 
OF    LOGARITHMIC   FUNCTIONS. 

1.   Required  to   develop   log.    (a  -\-  x)   by  Maclaurin's 
Theorem,  according  to  the  ascending  powers  of  x. 

Let    y   =  log.  (a  +  x)  .-.      (y)      =  log.  a, 
dy  l 


dx         a  +  x  \dx 

dzy  1 


dx2 

d  x3         (a  -+-  x)  \dx3/         a3 

d*y  2  .  3 

dx*~          (a  +  x)4 

.      X  X2  X3  X* 

.:  log.  ( a  4-  x )  =  log.  a  -\ h ,  etc. 

'a       2«2^3a3        4a^    ' 

2.   From   the    above   development,  required   to    deter 
mine  the  Log.  of  the  number  11. 

log.  (10  +  1)  =  log.  10  +  - l- — h  — +>  etc. 

~10        2.10 2    "^3.10 3        4.10-1 

log.  11  =  2,30258  +  I  —  —  +  -  -  -L,  etc. 

1    10        200    '    3000        40000    ' 

=  2,39788 
.-.,2,39788  -^  2,30258  =  1,04139.    Ans. 

The  algebraic  addition  of  six  terms  only  of  the  above 
series  for  log.  11,  is  sufficient  for  determining  the  hyp.  log. 


CALCULATION   OF   LOGAEITHMS.  187 

correctly  to  five  places  of  decimals.  This  result,  divided 
by  the  reciprocal  of  the  modulus  of  the  common  system, 
gives  the  common  logarithm  of  11. 

If  a  be  quite  large,  as  4000,  or  5000,  the  addition  of  only 

two  terms,  viz.,  log.  5000  and  ---    the   dif.  coef.   of  log. 

(5000  -f-  1)>  gives  log.  5001,  accurately  to  five  places  of 
decimals. 

We  can  never  call  the  "  differences  "  between  the  log 
arithm  of  one  number,  and  that  of  a  number  greater  by  1, 
the  differential  of  the  logarithm  of  that  number  ;  the  dif 
ferential  of  a  logarithmic  function  is  0,  and  not  1. 

3.  Required  the  development  of  y  =  log.  (1  +  »)  for 
B  =  l. 

Ans.     y  =  log.  2  =  1  —  -  +  -—  -  +  -—-+,  etc. 

2  ~3         4    '    5         6    ' 

The  summation  of  this  series  would  serve  to  give  the 
hyp.  log.  of  2,  were  it  not  of  such  slow  convergency  as  to 
render  a  correct  sum  for  six  places  exceedingly  laborious. 

4.  Required  Log.  2  from  the  development  ofy  =  log. 

—  ,  which,  when  x=  -,  becomes  evidently  log.  2. 
1  —  x  3 

.-.  (y)  =  log.  1  =  0; 
*  2  *     * 


dx        (1  —  z2)2    *    1—  x"   \dx  1  — 


, 


(I-*2)2       \dx* 

S3 

==  4'  etc'  ; 


188  DIFFERENTIAL  CALCULUS. 

We  have,  then,  if  x  =  - , 

3 

2x      =  0,66666666 

\x*    =  2469134 

o 

|aj5     —  164614 

?j-xi     =  13064 

^x*    =  1128 


-«»   =  102 


.-.  log.  2  =  ,69314708 

Now  ,69314708  -7-  2,30258  zn  ,30103  /.  ,30103  is  Log.  2. 

Having  obtained  Log.  11  =  1,04139,  and  Log.  2  = 
,30103,  we  find 

1,04139  +  ,30103  —  L.  22  —  1,34242 
1,34242  +1  =  L.  220  —  2,34242 

1,04139  —  ,30103  =  L.  5^  =  0,74036 
1,04139  X  2:r=L.ll2:=L.  121  —2,08278 
OJ4036  +1  =  L.  55  =  1,74036 

1,74036  X   2  =  L.  552  =  L.  2925    =  3,48072,  etc. 

When  we  have  obtained  the  logarithms  of  the  prime 
numbers,  we  easily,  as  above,  obtain  the  logarithms  of  all 
other  numbers. 

5.  Required  to  differentiate  y  —  x  log.  a?,  =  log.  xx. 

dx 

Ans.     dy  =  dx\og.  x-{-x  X —  =  dx\Qg.x-\-dx. 

X 

6.  Required  to  differentiate  y  =  log.  x2  =  2  log.  x. 

2dx 

Ans.    dy= . 

X 


LOGARITHMIC   FUNCTIONS.  189 

7.   Required  to  differentiate  y  =  log.  x  -\-  log.  (a  -f-  x)  = 

log.  [a  X  («  +  a)  ]  =  log.  (a  a  +  a2). 

(a  +  2*)  da; 
Ans.     a  v  = 


8.   Required  to  differentiate  y  =  log.  2  jc. 

2  log.  a;  X  dx 


._,   .  . 

Making  log.  x  ==.  z,  dy  =2  2  z  dz  =  2  log.  ic  <#g= 

9.  From  log.  y  =  £C,  to  find  c?y. 

Ans.      —  =  dx  .\  dy  =  ydx. 

10.  From  a  =  log.  £cy,  to  find  dy. 

a  ad  log.  a;  a  dx 

,  ,'.1/  =  -  /.  dv  =  --  ,  =  --  —  — 

log.  x  log.  2  x  x  log.  2  a; 

11.  Required  to  differentiate  y  =  (log.)2  a;,  by  which 
is  intended  log.  log.  a?. 

Putting  z  for  log.  JB,  c?  y  =  —  ,  but  d  z  =.  d  log.  x 

dx 

.-.  dy  =  -  --  . 

x  log.  * 

12.  Required  the  value  of  the  following  vanishing  frac 
tion,  when  05  =  1,  or,  which  is  the  same,  the  value  of  the 
difference  of  the  two  functions  of  a?,  each  of  which  be 
comes  co  when  x  =  1,  viz.  : 

x  log.  x  —  (x  —  1)  x  1 


(x  —  1)  log.  x  x  —  1        log.  x 

l_ 

p1        log.  x  -f- 1  —  1     p" 
Art.  171.         —  - — •  -\nr,  ~  \   „ i »         ^^ 


-  + 

X  X 


190  DIFFERENTIAL   CALCULUS. 

13.   Required  the  value  of  °B'X    *  °S'X  when  x  =  1. 

(log.*)2 

Ans.  — 1. 


204.  The  differentiation  of  an  algebraic  function,  which 
is  resolvable  into  factors,  may  be  much  facilitated  by  first 
taking  its  logarithm,  and  then  differentiating,  it  being  ob 
served  that  the  differential  of  a  logarithmic  function  may 
not  contain  a  logarithm,  but  be  purely  algebraic.  If  the 
function  is  not  algebraic,  and  is  not  resolvable  into  factors, 
this  method  may  be  used,  but  without  advantage. 

14.    Required  —  of  y  =  (a  +  a;2)  3  V«- 

(I  X 

We  have          log.  y  =  3  log.  (a  -\-  x  2)  +  —  log.  x 

dy        Sxdx        dx 
=  —  =  —     —  --  •  (1.) 

y         a  +  x2    '    2* 


after  substituting  the  value  of  y  for  y  in  equation  (1.) 

dy 


15.   Required--  of  y  —  x  (a2  +  a;2)  V«2  —  a2, 

d  x 

/.  log.  y  =  log.  x  +  log.  (a2  -f-  a2)  +  -  log.  (a2  — 
dy        d  x    ,      Ixdx  xd  x 


dy 


MAXIMA,    ETC.,   BY  LOGARITHMS.  191 

205.  Since  the  logarithm  of  a  quantity  or  function 
becomes  greater  as  the  quantity  becomes  greater,  and  less 
as  the  quantity  becomes  less,  the  maximum  of  the  quantity 
occurs  at  the  same  value  of  the  variable,  as  the  maximum 
of  its  logarithm  occurs.  We  may  avail  ourselves  of  this 
principle. 

Problem  33,  on  page  88,  rendered  in  more  general 
terms,  becomes  : 

16.  Required  to  divide  the  number  a  into  two  such 
parts  that  the  mih  power  of  one  part  multiplied  by  the 
nth  power  of  the  other,  shall  be  a  maximum. 

Let  x  =  one  of  the  parts,  and  y  =.  the  product  in  ques 
tion  ;  then, 


let  u  =  log.  y  —  m  log.  x  -\-  n  log.  (a  —  a;), 

mdx         ndx         dy 

then  du  —  d  log.  t/  =  —  —  =  — 

x  a—x          y 

du        d  log.  y        my          ny  dy 


d  x  d  x  x          a  —  x        y  d  x 

Whence,  if     —  =  0,  or  — *-  =  0,  we  have  x  =  -^-. 

d  x  y  d  x  m-{-n 

We  find  that  we  thus  eliminate  all  necessity  of  substituting 
the  value  of  y  in  —  ,  or  — — . 

d  x  y  d  x 

We  will  now  prove  that  we  need  not  substitute  the 
value  of  y  in  the  second  or  any  succeeding  differentiation 

d  i{ 

of  — ,  so  far  as  determining  maxima  is  concerned. 


192  DIFFERENTIAL    CALCULUS. 

,,r     ,  ydu        dy 

We  have  -  =  —  , 

d  x          d  x 

d  x  (dy  du  +y  d*  u)         d9  y 
dx*  ~~d** 


d  u 

therefore,  when  —  =  0, 

d  x 


In  such  use  y  being  the  quantity  of  which  the  logarithm 

dz  u 
is  taken,  must  be  positive  ;   hence  the  sign  of  -  will 


agree  with  the  sign  of  —  -  in  general,  y  being  a  common 
divisor  in  all  dif.  coefs.  of  w,  or  a  common  factor  in  all  dif. 
coefs.  of  y. 

In  the  particular  case,  u  and  y  are  maxima  in  necessary 


concurrence. 


306.  It  is  required  to  develop  y  =  log.  (x  -\-  h)  by 
Taylor's  Theorem,  according  to  the  powers  of  A,  the  ex 
pression  log.  being  general,  —  being  the  modulus. 


d*y  _ 
d 

d3 


EXPONENTIAL    FUNCTIONS.  193 


SECTION  XXVI. 

EXAMPLES  OF  THE  DIFFERENTIATION  AND  ANALYSIS 
OF  EXPONENTIAL  FUNCTIONS;  INCLUDING  EXAM 
PLES  FROM  COMPOUND  INTEREST,  AND  INCREASE 
OF  POPULATION. 

1.  From  a  =  bx  to  find  &,  where  a  and  b  are  any  num 
bers,  or  numerical  quantities : 

log.  a          L.  a 
log.  a  =  x  log.  b  .-.  x  =  - — -  =  — . 

log.  0  L.  o 

2.  From  a^  =  £*  to  find  — . 

d  x 

d  y        log.  b 

y  log.  a  =  x  log.  b.:—  = . 

d  x        log.  a 

3.  From  a  =  bmx  to  find  x. 

log.  a 

log.  a  =  m  x  log.  5  .-.  x  =  -     — . 

w  log.  6 

4.  From  a  =3  #— *  to  find  x. 

log.  a 


log.  a  =  —  x  log.  b .:  x  —  — 


log.  6 


From  this  result  we  may  infer  that  either  a  or  b  must  be 
less  than  1,  or  a;  must  be  held  to  be  of  a  value  opposite  to 
that  expressed  by  the  sign  in  the  function  (Art.  97). 

5.  From  a  —  bx  to  find  log.  x. 

log.  a  =  x  log.  J  /.  log.  x  =  log. 2  a  —  log. 2  5. 
17 


194  DIFFERENTIAL  CALCULUS. 

6.  From  a  =  - —  to  find  x. 

Pm- 

log.  a  =  x  log.  b  —  ra  x  log.  JP, 

log.  a 
log.  b  —  m  log.p 

7.  From  y  =  fL.  to  find  — , 

6X  d  x 

log.  y  =  sc  log.  a  —  a  log.  £>, 
df  log.  y  :=  —  =  (log.  a —  log.  b)  dx, 

.'.  —  =  -^  (log-  a  —  log  b). 

8.  From  y  =  a  x  log.  x  to  find  — . 

Let            z  •==.  log.  tc,  then  log.  y  =  x  log.  a  -f~  1< 
.-.  d  log.  y  =  —  —  d  x  log.  a  -| ; 


now 


2          log.  £ 

•••;£=  (log.  a+^a' log.*. 

9.  From  a  =  5log- x  to  find  x. 

log.  a  =  log.  x  log.  #, 

log.  a         L.  a 

•••1°8-!B=sp-=£r»; 

jc  is  therefore  the  natural  number  of  which  this  quotient, 

-  is  the  logarithm. 
L.  6 


EXPONENTIAL  FUNCTIONS.  195 


10.   From  y  —  ax  bnx  to  find  —  . 

d  x 

log.  y  =  x  log.  a  -f-  n  x  log.  b, 
d  log.  y  =  —  =  log.  aX  dx  +  n  log.  6  X 

.-.  —  =  a*  bnx  (log.  a  +  7i  log.  5). 


11.   From  a  a;  =  &*"  to  find—  . 


d  y  1  —  log.  a  +  log.  x 
-  —  -  —  -  - 
d  x  x2  log.  6 


12.   From  y  =  xx  to  find—. 

rf  X 


rf  y 

—  =  log.  x  d  x  -f-  d  x, 
dy 


=  !E«(log.fl!  +  l). 

13.  Required  the  value  of—      -  when  x  =  0. 

(Art.  171.)     p  =  log.  a  .  ax  —  log.  b  .  bx. 
?'=!, 

.*.  —  =  log.  a  —  log.  b  =  log.  — ,    Ans. 

qt  b 

x a 

14.  Required  the  value  of  —      —  when  x  =  a. 

log.  a -log.  a; 

Ans.     (1  —  log.  a)  a 


196  DIFFERENTIAL   CALCULUS. 

207.  The  successive  terms  of  a  series  called  progression 
by  quotient,  but  as  truthfully  progression  by  factor,  differ 
in  their  expression  by  a  varying  index  only. 

If  a  be  the  first  term  of  such  series,  and  q  the  factor, 
which,  by  association  with  a  produces  the  second,  q*,  <?3, 
etc.,  the  factors  which  produce  the  succeeding  terms,  we 
have,  after  putting  q  °  =  1  with  a  for  the  first  term, 

aq°,  aq,  aq\  aq*,  aq4,  etc., 

as  an  instance  of  the  successive  terms  of  such  series.  This 
may  be  called  the  general  form.  Any  one  of  these  terms 
is  determined  by  a  the  first  term,  q  the  common  ratio  and 
n  its  index. 

The  compound  interest  of  a  sum  of  money  for  a  term  of 
time,  the  interest  being  supposed  to  be  added  to  the  prin 
cipal  at  the  end  of  each  year,  is  represented  by  a  term  in 
such  a  series ;  and  all  its  terms  are  the  amounts  as  they 
are  constituted  at  the  end  of  each  year. 

For  the  better  understanding  of  this,  let  us  resolve  q 
into  1  +  — ,  where  1  represents  1  dollar  put  at  compound 

interest,  and  r  the  rate  per  cent. ;  consequently  —  repre 
sents  its  interest  for  one  year  in  the  proper  fraction  of  a 
dollar,  and  1  -f-  —  constitutes  the  amount  of  the  principal 
1  dollar  and  its  interest  at  the  end  of  the  first  year.  The 
quantity,  a,  is  any  sum  of  dollars,  and  is  constant  through 
all  the  terms ;  for  all  dollars  at  compound  interest  sever 
ally  are  like  the  1  dollar  mentioned  as  principal.  The 
amount  of  the  1  dollar  for  the  second  year  is  evidently 

r  \  2  /•  ~   \  3 

-) \  ,  for  the  third,  etc.,  f  1  -| j  ,  etc.,  and  for  any 

sum  a,  the  preceding  expressions  become  a  (l  -{-  —  J  , 
a(l4-  — 


PROBLEMS   WITH   VARIABLE   EXPONENTS.  197 

In  the  succeeding  context,  we  will  use  the  small  Roman 
r  instead  of  the  small  italic  r ;  as, 

r 
~  100  ' 

but  must  return  to  the  Italic  r  when  we  mention  rate 
per  cent.,  for  then,  in  a  distinct  sense  we  make  an  integer 
of  each  unit  of  r. 

We  have,  then,  for  successive  amounts  of  the  compound 
interest  on  any  sum  of  dollars,  a,  at  the  end  of  succes 
sive  years, 


the  sum  put  at  interest,  however,  for  the  first  year  is  enti 
tled  to  the  expression 


so  that,  as  a  series  which  may  ever  need  to  be  summed,  the 
term  having  the  index  n,  is  the  (n  -f-  1)  st  term. 

If  we  call  A  the  amount  of  principal  and  interest  of  a  sum 
at  compound  interest  for  n  years,  which  amount  is  nothing 
more  than  just  that  term  of  the  series  of  which  n  is  the 
index,  we  have  the  formulas  : 

.4=a(l+r)»,  (I.) 


L.A-L.a 


(4.) 
(5.) 


198  DIFFERENTIAL   CALCULUS. 

where  iS  represents  the  sum  of  n  terms  of  the  series,  a  q  °, 
«<?,  a<?2,  etc.  ;  for  the  demonstration  of  formula  (5.),  we 
must  refer  to  a  complete  treatise  of  algebra,  since  it  is  not 
like  (2.),  (3.),  and  (4.)  deduced  from  (1.) 

We  are  furnished  now  with  the  means  of  resolving 
questions  which  make  n  to  be  #,  or  a  variable,  and  any 
of  the  other  quantities  to  be  y. 

15.  If  342  (a)  dollars  be  put  at  compound  interest  at 
5  ,  (r)  per  cent.,  required  how  the  amount  (A)  is  increasing, 
compared  with  the  years,  at  the  end  of  3  J-  years. 

From  y  =  a  (1  +  r)  * 

/.=alog.(l+r)  (1+r)3* 


=  342  Log.  (m\    /^     X  2,30258  =  20,  Ans. 
dx  \m)   \iooj 

Hence  it  is  increasing  20  times  as  fast  as  the  years. 

16.  The  sum  of  1200  (a)  dollars  was  put  at  compound 
interest  till  the  amount  (A)  accrued  to  be  2525,82  dollars. 
If  we  first  assume  the  number  of  years  to  have  been  11, 
and  immediately  proceed  to  consider  them  more,  how  is 
the  implied  rate  per  cent,  disposed  to  change,  in  accord 
ance  with  the  assumed  variation  of  time  ? 

On  examining,  in  formula  (4.),  ^,  we  observe  that  we 
have  to  do  only  with  the  value  of  this  fraction,  which,  in 
the  present  problem,  is  2,1048.  We  must  remember  the 
denominator  of  ris  100. 

.-.r=100  X  (2.1048)  ir—  100, 
or,  y  =  100  X  (2.1048)  r  —  100. 


PROBLEMS   WITH    VARIABLE    EXPONENTS.  199 

.-.  ^  =  100  X  2.30258  X  L.  2.1048  X  (2.1048)  F  X  (—  j,) 

230,258  X  0.32322  X  1,07 
—  -  _  ---  .        _  —  -  —  56583,  Ans. 

Here  we  are  obliged  to  remember  the  differential  of  the 
variable  exponent. 

Hence  the  rate  per  cent.,  which  happens  to  be  7,  is  dis 
posed  to  diminish  6583  ten  thousandths  of  one  per  cent. 
The  factor,  2.30258,  will  be  remembered  as  the  reciprocal 
of  the  modulus. 

17.  On  the  1st  day  of  January,  1864,  the  amount  of 
principal  and  interest  of  a  sum  of  money  having  been 
at  compound  interest,  at  7  (r)  per  cent.,  was  found  to  be 
3579  (a)  dollars;  required  to  find  the  number  of  years 
distant  before  or  after  that  date,  when  the  compound 
interest  of  the  sum,  whatever  it  may  have  been,  that  was 
originally  put  at  compound  interest,  should  be  found  in 
creasing  60  times  as  fast  as  the  years. 

With  reference  to  the  date  given,  the  amount  3579  dol 
lars,  is  not  the  A  for  the  proposed  investigation,  but  is  a,  the 
sum  considered  to  be  put  at  interest  with  reference  to  both 
future,  and,  as  it  were,  past  time.  If  n  in  the  formula 
should  be  negative,  then  A  as  less  than  a,  may  be  calcu 
lated  for  any  past  time.  Now  the  question  does  not  ask  for 
the  value  of  this  A,  but  for  its  rate  of  change,  which  was 
just  the  variation  of  the  interest  only.  Therefore,  calling 
this  Ay  or  the  interest  either,  y,  we  have 


—  a  (1  +  r)*  =  3579     ~ 
i 


...  H=60  =  3579  X  1UW268  X  L.  ©   O'. 


_  L.  60  —  L.  3579  —  L.  (L.  107  —  L.  100)  —  L.  2.30258 
L.  107  —  L.  100 


200  DIFFERENTIAL   CALCULUS. 

Now,  L.  107  =  2.02938 ;  L.  100  =  2  .-.  L.  107  —  L.  100  = 
.02938, .-.  —  L.  (L.  107  —  L.  100)  =  —  L.  .02938; 

L.     60  =  -f  1,  +  77815 

—  L.   3579  =  —  3,  —  55376 

—  L.  .02938  =  +  2,  —  46805 

—  L.  2.30258  =  —  0,  —  36222 

__  4,  _  38403 
_|_  3,  _|_  77815 


—  60578 

Now,  —,60578  -^  ,02938  —  —  20,62  =  —  (20  years,  7 
months,  13  days)  ;  and  the  date  desired  is  May  18th, 
1843,  Ans. 

18.  (a.)  If  a  body  be  put  in  motion  through  a  resisting 
medium,  by  a  force  which  impels  it  10  rods  in  the  first 
second  of  time,  9  rods  in  the  next  second,  and  so  on,  so 
that  in  any  second  of  time  the  distance  impelled  shall  be 
Y^ths  of  the  distance  in  the  preceding  second,  required 
how  far  it  will  go  in  all  time. 

With  reference  to  formula  (5.),  /8  is  the  distance  re 
quired,  a  is  10  rods,  and  q  is  -^  ;  now  q  being  less  than  1, 
its  infinite  power  is  0.  So  that 

8  —  —  10  -. TV  =  100  rods,  Ans. 

This  determines  a  limit  for  £;  for  the  function  — ^- — -, 

q  —  l 

can  have  no  mathematical  maximum,  as  by  definition. 

(b.)  Required  how  fast  the  body  is  moving  at  the  end 
of  16  seconds ;  i.  e.,  its  constant  rate,  as  it  were,  for  an 
infinitesimal  space  of  time,  but  mentioned  in  the  language 
of  rate  per  second. 


PROBLEMS  WITH  VARIABLE  EXPONENTS.     201 

Since  S,  the  whole  distance  attained  at  the  end  of  16 
seconds,  will  be  varying  just  as  the  velocity,  we  will 
substitute  y  as  velocity  for  S,  for  the  purpose  of  differen 
tiation,  16  or  the  number  of  seconds  being  x ;  then, 


—  _ioo(^     —100 


10 


£  =  -  100  X  2,30258  X  L.  (')  (i 

=  230,258  X  ,04576  X  ,1853 

=  1,9526  rods,  Ans. 

(c.)  Required  the  actual  distance  moved  through  in  the 
17th  second. 

19.  (a.)  A  man's  property,  on  January  1,  1850,  consisted 
of  an  investment  of  1500  (a)  dollars  in  stocks,  paying  an 
annual  interest  of  7  (r)  per  cent.,  but  which  is  to  remain  in 
vested.  The  remainder  of  his  property  was  unemployed, 
with  reference  to  producing  any  income,  but  was  salable  at 
the  date  mentioned  for  2700  (#/)  dollars,  and  was  destined 
to  depreciate  at  the  rate  of  4  (r')  per  cent,  annually,  as 
indeed  it  had  been  previously.  Required  the  date  of  the 
least  value  of  the  general  balance  of  his  property. 

Let  y  =  the  sum  required,  =  A  -f-  A'^  and  x  =  the 
number  of  years'  difference  of  date.  Then  by  the  formula, 

y  =  a  (1  +  r)*  +  «'(!—  r')x 


/.  when  y  is  a  minimum,  for  we  know  from  logical  consid 
erations  that  y  has  only  a  minimum,  we  have  : 


202  DIFFERENTIAL   CALCULUS. 


/  1+r  x  *  _  —  o  'log.  (1—rQ 
\1  —  r'/    "    Tlog.  (1  +  r)      ' 

—  a'L.(l  —  r') 


aL.  (1  +  r) 

__L.a'—(Log.)«  (l-r')-L.  a-(Log.)2  (1  +  r)  ^ 
L.(l  +  r  )  —  L.  (1-rO 

where         -  (Log.)2  (1  —  r')  =  L.  (—  L.  (1  —  r'); 

and  is  therefore  a  positive  quantity,  L.  (1  —  r  ')  being  neg 
ative,  and  —  L.  (1  —  r  ')  being  positive,  and  in  real  arith 
metical  expression,  being  without  the  —  sign. 

The  advantage  of  electing  to  place  the  negative  sign 
before  L.  (1  —  r'),  in  preference  to  any  other  factor,  is 
manifest. 

(#.)  Required  the  date  at  which  the  two  species  of 
property  become  of  equal  values  ;  and  also  that  value. 

20.  (a.)  A  certain  country  consists  of  two  districts,  East 
ern  and  Western.     On  a  certain  date  the  Eastern  contained 
6,272,000  inhabitants,  who  were,  and  had  been  increasing 
at  the  rate  of  16  per  cent,  in  10  years.     The  Western  con 
tained  at  the  same  date  9,035,000  inhabitants,  who  were 
and  had  been  decreasing  at  the  rate  of  5  per  cent,  in  8 
years.     Required  the  different  date  before  or  after,  of  the 
minimum  population  of  the  country. 

(b.)  Required  the  date  of  like  population  of  the  dis 
tricts. 

(c.)  Required  the  date  at  which  the  Eastern  district 
must  contain  nine  times  as  many  inhabitants  as  the 
Western. 

21.  Required  the  number  by  which,  if  we  divide  a  and 
raise  the  quotient  to  the  power  indicated  by  that  divisor, 
the  power  shall  be  a  maximum. 


PROBLEMS   WITH   VARIABLE  EXPONENTS.  203 

Let  y  =  the  power ; 

then  y  = 

•.  log.  y  —  &  log.  —  =  x  log.  a  —  x  log.  «, 

rfy 

•'•  —  =  y  log.  a  —  y  —  y  log.  a, 

ot  <f  y 

.•.  log.  —  =.  1  when  —  =0, 

x  d  x 

.-.  L.  -  =  .43429, 

.-.  L.  x  —  L.  a  —  .43429, 
L.  ic  =  L.  a  —  L.  2.71828, 

a 
~~  2.71828  ' 

208.  In  the  common  arithmetical  computation  of  com 
pound  interest  for  cases  when  there  are  months  and  days, 
additional  to  entire  years  as  the  time,  the  usual  direction 
is  to  find  first  the  amount  of  principal  and  interest  for  the 
entire  years,  on  which,  as  principal,  to  compute  the  interest 
for  such  additional  months  and  days.  But  this  course 
will  always  give  the  entire  compound  interest  somewhat 
too  great,  because  it  assumes  that  the  interest  is  to  accu 
mulate  uniformly  during  such  months  and  days  by  a  rate 
that  is  directly  proportional  to  the  result  that  would  accrue 
for  an  additional  year. 

When  a  sum  is  put  at  simple  interest  for  one  year  at  a 
given  rate  per  cent.,  the  virtual  amount  of  principal  and 
interest  is  greater,  relatively  to  the  time,  during  the  later 
or  last  months,  than  during  the  first  and  earlier  months, 
for  the  reason  that  the  unpaid  interest  of  the  early  month 


204  DIFFERENTIAL   CALCULUS. 

is  itself  on  interest  during  the  later  month.  The  idea  of 
calling  the  use  of  money  worth  a  given  per  cent,  for  a 
year,  is  therefore  a  compromise  for  ready  convenience. 

Let  us  now  actually  compute  the  true  interest  of  100 
dollars  for  the  successive  months  of  a  year,  at  six  per  cent, 
per  year  by  reducing  n  in  the  formula  (1.)  p.  197  to  months, 
which  is  done  by  giving  to  it  the  numerator  12.  Formula 

/» 

4th  then  becomes,  when  r  is — ,  so  far  as  the  constitution 

100 ' 

of  A  is  concerned,  or  the  amount  for  one  year,  but  still 
remains  as  r,  for  the  first  rnontKs  rate. 


whence,  when  n  =.  1,  we  have 

r  =  ,00487  ;  and  r  =  ,487  ; 

a  decimal  fraction,  of  which  the  unit  would  be  1  per  cent. ; 
or  it  may  be  read  48  cents  7  mills,  as  the  interest  of  100 
dollars  for  1  month,  at  nominal  six  per  cent.,  and  indeed 
virtual  six  per  cent,  for  1  year  by  the  language  of  com 
promise.  If  we  were  to  speak  of  a  mathematical  per  cent., 
then  multiplying  ,487  by  12,  we  arrive  at  5T8Q%4g-  as  the 
mathematical  per  cent,  for  the  rate  of  interest  of  money 
for  the  first  month,  when  at  "  interest  at  six  per  cent,  the 
year." 

Calling  n  —  2,  and  obtaining  another  result  for  r,  and 
subtracting  that  obtained,  we  have  for  the  interest  of  100 
dollars  for  the  second  month,  49  cents.  Continuing  in  this 
manner,  we  find,  as  the  result,  the  interest  for  the  twelve 
successive  months  to  be,  in  cents,  48.7;  49.0;  49.2;  49.4; 
49.6;  49.8;  50.1;  50.3;  50.5;  50.8;  51.1;  51.4,  which, 
added,  make  600  cents,  or  6  dollars. 


PROBLEMS   WITH    VARIABLE   EXPONENTS.  205 

If  n  be  considered  to  have  values,  then,  between 
integral  numbers  for  years,  or  to  be  a  quantity  having 
"  flowing  "  values,  we  are  enabled  to  derive  true  computa 
tions  for  compound  interest  for  other  lengths  of  time  than 
entire  years.  In  this  consists  the  value  of  that  formula, 
and  of  this  analysis  afforded  by  the  differentiation  of  expo 
nentials. 

22.  Required  the  true  time  in  which  a  sum  of  money 
becomes  doubled  when  put  at  compound  interest,  at  5  per 
cent.,  and  how  much  more  it  is  than  by  the  arithmetical 
way. 

Axis.    14  yrs.  2  mo.  15  da.,  being  2  da.  more. 

23.  The  white   population  of  the  United  States,  from 
June    1,  1830,  to   June    1,   1840,  increased  34  per  cent. 
Required  its  annual  ratio  of  increase,  and  in  what  time  it 
must  have  become  doubled. 

Ans.   2T9^63  per  cent.   Doubled  in  23  yrs.  8  mo.  12  days. 


24.  Required  the  true  compound  interest  of  360  dollars 
for  5  years,  6  months,  and  24  days,  and  how  much  less  the 
true  is  than  the  usual  arithmetical  result. 

Ans.     $137.80.     The  difference,  34  cents  less. 

A  true  but  impracticable  definition  of  compound  interest, 
ignoring  the  termination  of  entire  years,  as  essential,  may 
be  :  A  sum  of  money  is  said  to  be  put  at  compound  inter 
est  for  a  period  of  time,  when  the  value  of  the  use  of  a 
unit  of  it  for  a  time  however  short,  but  definitely  stated, 
is  agreed  upon  at  a  rate  (or  proportion  of  that  unit),  and 
such  rate  or  interest  is  added  to  that  unit  at  the  end  of  such 
period,  and  the  use  of  the  sum  of  them  for  a  similar  period 
of  time  is  estimated  at  the  same  rate;  and  this  second 
interest  is  added  to  the  sum  mentioned  as  the  sum  for  use 
during  a  third  similar  period  of  time,  and  so  on  to  the  end 
18 


206  DIFFERENTIAL   CALCULUS. 

of  the  last  similar  period  of  time,  and  the  formation  of  a 
final  sum,  called  amount  of  principal  and  the  interest. 


SECTION  XXVII. 

DIFFERENTIATION   OF   CIRCULAR  FUNCTIONS. 

[Unlike  all  the  previous  sections,  the  present  presup 
poses  the  principles  of  Analytical  Trigonometry  to  be  un 
derstood.] 

209,  A  circular  independent  variable  may  be  an  arc  of 
a  circle,  or  its  sine,  tangent,  or  other  trigonometric  line 
referred  to  the  arc. 

A  circular  function  is  a  function  of  stfme  linear  trigo 
nometric  variable ;  it  may  be  the  arc  itself,  when  the  arc  is 
not  assumed  as  independent  variable ;  it  may  be  a  sine, 
tangent,  etc.,  of  the  variable  arc. 

210,  Trigonometrical  quantities  are  all  to  be  considered 
numerical    linear    amounts   in   their   result.       Quantities 
strictly  algebraic,  as  factors,  etc.,  may  contribute  to  this 
result ;  as,  m  in  sin.  ra  ic,  n  in  n  tan.  x.     These  quantities, 
when  they  are  powers,  and  when  their  amounts  agree  with 
certain  areas,  are  nevertheless  to   be  regarded  as  linear 
amounts,  or  multiplications  of  a  line.     Hence  trigonomet 
rical  quantities  are  special  in  kind,  and  not  general,  like 
arithmetical  or  algebraic  quantities. 

When  an  arc  is  made  variable,  we  may  call  it  the  arc  x. 
If  the  radius  of  the  circle  be  1,  or  not,  its  variable  arc  is  of 
unlimited  length,  by  repetitions  of  itself  if  need  be.  All 
the  principles  of  trigonometry,  and  of  the  differentiation  of 


CIRCULAR  FUNCTIONS.  207 

circular  functions,  are  intended  to  apply  to  this  unlimited 
arc. 

The  expressions  for  circular  functions,  sin.  cc,  tan.  te,  etc., 
are  intended  to  signify  the  sine,  tangent,  etc.,  of  the  arc 
which  is,  in  length,  x  of  the  units  of  which  the  radius  is  1, 
unless  otherwise  expressed.  We  can  hardly  call  x  sin.  a 
a  circular  variable  function,  the  variable  ar,  not  being  re 
stricted  to  the  circle. 

It  is  necessary  to  make  the  interpretation  of  the  whole 
expression  intended  as  the  circular  function,  however  the 
variable  x  may  occur  in  it,  contribute  to  a  homogeneous 
result.  In  x  sin.  a?,  for  instance,  the  prefixed  x,  being  a 
factor  to  sin.  x,  must  be  abstract  numerical,  but  of  the 
same  numerical  value  as  that  of  x  in  sin.  ic,  so  that  the  result 
is  a  linear  amount.  In  x  -\-  sin.  x  it  is  necessary  to  inter 
pret  the  isolated  -x  as  the  arc  to  which  another  linear 
amount,  sin.  ic,  is  added. 

Powers  of  the  sine  of  the  arc  a;,  cosine  of  the  arc  cc,  etc., 
are  expressed  by  the  exponent  attached  to  the  prefix,  as, 
sin.2  ce,  cosec.n  x.  This  leaves  a  distinctive  signification 
for  sin.  a;2,  etc.,  which  is  obvious,  and  requires  no  use  of 
parentheses. 

In  the  expression,  sin.  a;,  it  is  scarcely  necessary  to 
remark  that  the  prefix  sin.  is  not  mathematically  separable 
from  #,  and  we  will  not  adopt  that  inverse  notation  which 
from  y  =  sin."1  x  would  attempt  to  derive  sin.  y^=x. 
Indeed,  we  are  already  committed  to  regarding  sin."1  x  as 

equivalent  to  — 

sm.  x 

In  the  function  x cos- x  the  exponent  must  be  taken  in  its 
numerical  sense,  apart  from  being  linear,  and  the  root  x 
must  be  the  same  arc  of  which  the  cosine  is  intended, 
raised  in  its  numerical  sense  to  the  power  cos.  x\  the 
resulting  power  may  at  last  be  taken  as  that  of  the  arc. 

The  expression  log.  sin.  x  must  be  abstract  numerical. 


DIFFERENTIAL   CALCULUS. 

The  expression  sin.  log.  x  must  be  linear,  and  the  arc 
intended  must  be  of  the  numerical  length,  log.  a?,  radius 
being  1, 

211.  It  is  evident  that  the  differential  of  an  arc  of  the 
length  zero,  may  be  called  identical  with  the  differential 
of  the  tangent  of  it,  or  chord  of  it. 

We  are  to  understand  sin.  (sin.  x)  to  signify  the  sine  of 
the  arc  which  is  of  the  length  sin.  33,  which  is  of  course 
a  less  arc  than  x  of  the  same  circle.  As  with  logarithmic 
functions  we  use  (log.)  2  x  for  log.  (log.  &),  so  we  will  use 
(sin.)2  for  sin.  (sin.  x).  The  second  power  of  sin.  x  will  be 
sin.2  x  without  the  parenthesis. 

Such  an  expression  as  asiu-x  may  be  called  an  exponen 
tial  circular  function. 

Circular,  Logarithmic,  and  Exponential  Functions,  are 
called  Transcendental  Functions. 

212.  In  order  to  differentiate  sin.  tc,  we  have  for  radius  1, 
if  a  be  any  arc,  and  b  be  any  additional  arc,  by  the  ratio  of 
corresponding  parts  of  similar  right-angled  plane  triangles: 

sin.  (a  +  b)  —  sin.  a  :  tan.  b  : :  cos.  a  :  1 ; 
that  is, 

sin.  (x  -\-  h)  .  —  sin.  x  :  tan.  h  : :  cos.  x  :  1 ; 

but  if  h  =  0,  sin.  (x  -f-  h}  —  sin.  x  becomes  d  sin.  cc,  and 
tan.  h  becomes  d  ic,  or  differential  of  the  arc  x. 

.'.  d  sin.  x  :  d  x  : :  cos.  x  :  1 
.*.  d  sin.  x  =.  cos.  x  d  x. 

213.  If  the  arc  be  designated  otherwise  than  by  cc,  as 
for  instance  by  ccn,  or  x  -\-  £c3,  etc.,  then,  instead  of  d  a?,  we 
must  substitute  the  differential  of  that  algebraic  or  other 
function  of  ic,  which  does  designate  the  intended  arc. 


CIRCULAR   FUNCTIONS.  209 

214.  If,  moreover,  the  function  to  be  differentiated  be 
some  function  of  sin.  a*,  as  sin.2  ic,  an  instance  of  which 
will  immediately  follow,  we  must  differentiate  it  as  any 
algebraic  power,  and  make  the  differential  of  the  root  a 
factor  in  the  differential  required. 

215.  In  order  to  differentiate  cos.  x  we  have 

cos.  x  =n  (1  —  sin.2  x)  *, 

/.  d  cos.  x  =  (d  (1  —  sin.2  x)  *), 

=  —  £  (1  —  sin. 2  x)  ~  *  X  2  sin.  x  cos.  x  d  cc, 

sin.  x  cos.  x 
~  (1  —  sin.2*)*' 
sin.  x  cos.  <£ 

= d:  #  =  —  sin.  x  d  x. 

cos.  # 

216.  In  oraer  to  differentiate  tan.  x  we  have 

,  sin.  x         cos.  x  d  sin.  x  —  sin.  x  d  cos.  x 

d  tan.  x  =  d  -    -  = 

COS.    £  COS.2  £ 

cos. z  x  -{•  sin. 2  a: 

/.  d  tan.  a  =  —  -  d  x  ; 

cos.2  a; 

but  cos.2  x  +  sin.2  x  =  1 

.*.  d  tan.  jc  = d  x  =  sec.2  x  d  x. 

cos.*x 

217.  In  order  to  differentiate  cot.  x  we  have 

,1  d  tan.  a; 

a  cot.  x  =  a = 

tan.  x  tan. 2  x 

sec.2  a; 

— d  x  =  —  cosec.  *  x  a  x. 

tan.2  a; 

18* 


210  DIFFERENTIAL   CALCULUS. 

218.  In  order  to  differentiate  sec.  x  we  have 

1  sin.  x 

a  sec.  x  —  a =  -     -  d  x 

cos.  x          cos.  2  a; 

tan.  x 
= d  x  =  tan.  a?  sec.  xa  x. 


219.  In  order  to  differentiate  cosec.  x  we  have 

d  cosec.  x  —  d = —  d  x 

sin.  x  sin.  a  x 

cos.  x  1 

= : X  — —  =  —  cot.  x  cosec.  x  a  x. 

sin.  x  sin. a; 

220.  Therefore,  by  recapitulation : 

d  sin.  x      =       cos.  x  d  x 
d  cos.  a;     =  —  sin.  x  d  x 
d  tan.  x     =       sec. 2  x  d  x 
d  cot.  x     =  —  cosec. 2  aj  <#  tc 
<£  sec.  x      =       tan.  35  sec.  x  d  x 
d  cosec.  x  =  —  cot.  x  cosec.  x  d  x. 

If  each  of  these  functions  be  y,  the  expressions  for  their 
first  dif.  coefs.  are  obvious. 

221.  Whenever  occasion  may  require  the  differentiation 
of  a  circular  function,  for  radius  R  other  than  1,  it  is  neces 
sary  to  employ  R  in  the  place  of  1  in  the  course  of  the 
method  of  determining  the  differential,  because,  although 
1  ==  I2,  this  would  not  be  true  of  R  and  R2. 

1.   Required  to  develop  sin.  x  by  Maclaurin's  Theorem  : 
y  =  sin.  x (y)  =  0 

•  •  •  •(£)='         . 


dy 

—  =.  COS.  CC. 

dx 


CIRCULAR   FUNCTIONS.  211 


„.=-«*.«. 


=  —  cos.  x  . 


=       sin.  x  .  .  (  — -    —  0 


dx* 

=       cos.  x ( - — —  1  =  1,  etc. 


sin.  x  =.  x 1 -K  etc. 

1.2.3    '     1.2.3.4.5        1.2.3.4.5.6.7    ] 

2.   Required  to  develop  cosine  x. 


y  =  cos.  x   ........  (y)    —  1 

dy  .  (dy\ 

-  =  —  sin.  cc  .......        _  )  —  0 

dx  \dxl 


/.  cos.  cc  =  l  ---  —  —  U  etc. 

1.2    '1.2.3.3          1.2.3.4.5.6    ' 

323.   If  we  take  the  expression  for  the  developed  sin. 
and  differentiate  it,  we  have 


, 

a  sin.  cc  =  a  x  ---  --  —  ,  etc., 

1.2      ~  1  .2.3.4 

=  (1  —  —  H  --  —  --  ,  etc.)  d  x 

1.2    r  1  .2.3.4 
=  cos.  x  d  as,  as  by  development  of  cos.  a?. 

323.   In  like  manner  if  we  take  the  expression  for  the 
developed  cos.  jc,  and  differentiate  it,  we  have 

x'3  xb 

d  cos.  x  =.  —  (x  -I  ------  1-,  etc..)  d  x 

r  1.2.3         1.2.3.4.5  n 

=  —  sin.  x  d  x  as  by  development  of  sin.  a;. 


212  DIFFERENTIAL   CALCULUS. 

The  summation  of  the  series  expressive  of  sin.  x  and 
cos.  a;,  for  particular  lengths  of  the  arc  a:,  must  give  the  nat 
ural  sine,  natural  cosine  of  such  arc.  The  sine  and  cosine 
of  one  arc  being  obtained,  the  sine  and  cosine  of  m  times 
such  arc  may  be  found  by  the  following  formulas,  of  which 
we  omit  the  demonstration  : 

m(m  —  l)(m  —  2) 

sm.  m  x  =  m  cos.m~~lx  sm.  x  --  -cos.m"3aj 

2i    •    O 

sin.3  x  -)-,  etc. 

m  (m  —  1) 

cos.  m  x  =  cos.m  x  --        —  cos.m"J  x  sin.  2  x  -f- 


2.3.4 


cos.m"4  x  sm.4  x  —  ,  etc. 


224:.  In  a  manner  similar  to  that  of  sin.  x  and  cos.  x, 
may  tan.  a?,  cot.  tc,  etc.,  be  developed.  Such  are  developments 
of  sin.  a;,  cos.  a;,  etc.,  depending  on  a  portion  x  of  the  arc 
as  assumed  variable,  useful  when  the  arc  is  known.  But 
we  may  equally  develop  the  arc  a?,  in  terms  of  some  func 
tion  of  it,  sin.  a?,  cos.  cc,  tan.  cc,  etc.,  and  in  doing  so,  while 
we  will  preserve  the  notation  as  already  used,  we  are  obliged 
to  regard  the  function  y  as  the  independent  variable,  and 

x  the  arc  as  the  dependent  variable ;  whence  -^ ,  in  such 
case,  becomes  the  reciprocal  of  those  inferred  from  Art. 

219   for  1*, 

d  x 

3.   Required  to  develop  x  in  y  =  tan.  x. 

y  •=.  tan.  x  /.  when  y  =  0     .     .   (x)  =  0 

d  x  _         1  1  Sdx\  

d  y         sec. 2  x         1  +  y  *  '  \d  y/ 


d*x  2y  /d*x\ 

wJ=  ° 


CIRCULAR  FUNCTIONS.  213 

d*x  2  Sd*x 


2  Sd*x\ 

=  ---  h  *  V«     .     .     .    (-      )=  — 

(1+y2)2  \dy*S 


dy 


—  =         ^  I 

<*y4       (i-t-y2)3 


2 


23.3 

;±*"y 


where  s,  /,  s"  are  quantities  factors  to  y, 
/.  x  ==  tan.  x  —  J  tan.3  tc  -f-  |  tan.5  a;  —  j-  tan.7  cc  -{->  etc. 
If  now  x  be  an  arc  of  45°,  tan.  x  =  1  =  radius, 
/.  arc  45°  =  1  —  $  +  -i  —  f  +  £  — ,  etc., 

which,  from  its  slow  convergency,  is  not  readily  summed. 
Its  sum  is,  in  terms  of  radius  =  1,  the  length  of  the  arc 
of  45°,  or  the  eighth  part  of  the  circumference  of  the  circle. 
By  the  aid  of  the  trigonometrical  formula, 

tan.  a  +  tan.  b 

tan.  (a  +  b)  =  -  — , 

1  —  tan.  a  tan.  b 

we  may  obtain  Euler's  series  for  the  same  purpose,  which 
is  much  more  convergent.  For  when  a  +  b  =  45°,  tan. 
(a  -f-  b)  =  1,  therefore 

tan.  a  -J-  tan.  b  =  1  —  tan.  a  tan.  b. 

If  now  either  tan.  a  or  tan.  b  were  given,  the  other  be 
comes  determinable  from  this  equation.  Thus,  if  we  sup 
pose 

1  1     ,  tan. b 

tan.  a  =  — .  then  — h  tan.  o  =  1 , 

n  n  n 

1  -f-  n  tan.  b  =  n  —  tan.  b .-.  tan.  b  =  — 

n+  1 


214  DIFFERENTIAL   CALCULUS. 

Developing  respectively  tan.  a  =  -  and  tan.  b  =  --  ,  by 

n  n  -\-  1 

the  method  last  found  for  tan.  x  we  have 

llll 
a  =  —  ----  -------  h  ,  etc. 

n        3n3    '    5n*        7  n?    ' 

n  —  1  n  —  1 

1  ----      —  k  etc. 

' 


n+l          3(n  +  l)3         5(n  +  l)5         7(n+l)7 


The  value  of  n  being  arbitrary  if  we  make  n  —  2,  for 
this  value  makes  the  two  series  converge  with*  a  near 
equality,  we  have,  if  a  sufficient  number  of  terms  be 
summed, 

4  (a  +  b)  =  45°  X  4  =  3.141592653589793, 

for  the  ratio  of  the  semi-circumference  of  a  circle  to  radius, 
or  of  the  whole  circumference  to  the  diameter. 


We  have  already  given  the  development  of  the 
sine  of  an  'arc  in  terms  of  the  arc.  If  it  be  desired  to  calcu 
late  numerically  the  natural  sine  of  an  arc  designated  by 
degrees,  minutes,  and  seconds,  as  for  instance  for  27°  10'  0", 
it  is  necessary  to  translate  this  designation  by  degrees,  etc., 
into  numerical  parts  of  radius  1.  Thus,  27°  10'  =  1630', 
and  180°  =  10800',  and  1\^ff  of  3.1415926  is  .47414777 
the  length  of  the  arc  of  which  the  natural  sine  is  re 
quired. 

If  now  we  select  for  use  .4741  as  the  arc,  and  sum  merely 
three  terms  of  the  development,  we  shall  have  the  usual 
tabular  amount  to  five  decimal  places  : 


nat.  sin.  27°  10'  =  .4741  —  —  +         -  —  ,  etc. 

2.3      '    2.3.4.5 

=  .45658 


CIRCULAR   FUNCTIONS.  215 

By  logarithms  we  have 

Log.  .4741 —  1 .  +  67587 

3 


Log.  .4741 3 —  1.  +  02761 

—  Log.  (2.3  =  6)    ....  .  —  77815 

Log.  .01776 -  2  .  +  24946 

Again, 

Log.  .4741 —  1 .  +  67587 

5 


Log.  .4751 5 —  2.  +  37935 

—  Log.  (2.3.4.5  =  120)      .      -  2  .  —  07918 

Log.  .0001996 —4. +  30017 

Now,  .47414 

—  .01776 


.45638 
+  .0001996 

.45658     nat.  sine  required. 

From  the  natural  sine  of  an  arc  we  easily  obtain  its 
natural  cosine,  natural  tangent,  cotangent,  secant,  and 
cosecant. 

4.  Required  to  differentiate  y  —  sin.  (a  +  m  x). 

Ans.     d  y  =  m  cos.  (a  +  m  x)  d  x. 

5.  Required  the  dif.  coef.  of  y  =  sin.  x  tan.  n  x. 

Ans.     —  =  cos.  x  tan.  n  x  +  n  sin.  x  sec.8  n  x. 
d  x 

6.  Required  dif.  coef.  of  y  —  sin.2  x. 

Ans.     —  =  2  sin.  x  cos.  x  =  sin.  2  x. 
d  x 


216  DIFFEEENTIAL   CALCULTTS. 

7.   Required  dif.  coef.  of  y  =  log.  sin.  x. 


dy         cos.  x 

Ans.     —  = = 


d  x         sin.  x         tan.  x 


d  y 

8.   From  sin.  x  —  cot.  y,  to  find  —  . 

d  x 

cos.  x  d  x  =  —  cosec.  ^  y  d  y 

dy  cos.  a; 


:=  —  cos.  x  sin. 


cosec.  2y 


9.  Required  the  value  of  —  '—  when  x  =  0. 

tan.z 

Ans.     1. 

10.  Required  value  of  -  -  when  x  =  0. 

sin.  x  3 

Ans.     £. 

11.  Required  the  values  of  or  when  sin.  a;  is  a  maximum 
and  minimum. 

Ans.     x  is  a  maximum  at  90°,  450°,  etc.,  and  a  minimum 
at  270°,  or  —  90°,  630°,  or  —  450°,  etc. 

12.  Required  the  value  of  ic,  when  y  =.  sin.  x  —  sin.2  x 
is  a  maximum. 

dy 
-  =  cos.  x  —  2  sin.  x  cos.  x  =  0, 

X 

/.  1  —  2  sin.  a  =  Q 
.•.1  =  2  sin.  tc, 
/.  oj  =  arc.  30°. 

13.  From  y  =  XCOB-X  to  find  ^. 

log.  y  —  cos.  cc  log.  JK, 

<2  y          cos.  x  d  x 

—  =  --  sm.  x  log.  x  d  x, 

y  x 


GEOMETRICAL  ILLUSTRATIONS.  217 


/.     y  =  g.cof.x-1  cogj  x  —  XCOB.X  sin<  x  IQ~  Xi 
dx 

14.  From  cot.  y  =  xco'-x  to  find  i?. 

d  y         (#  sin.  a;  log.  x  —  cos.  x)  cot.  y 

Ans.      -  =  -  — . 

dx  x  cosec.2  y 


SECTION  XXVIII. 

GEOMETRICAL  ILLUSTRATIONS  OF  THE  VALUES  OF 
FUNCTIONS,  AND  THE  CORRESPONDING  VALUES  OF 
THEIR  VARIABLES;  ALSO  OF  THE  VALUES  OF  DIF 
FERENTIAL  COEFFICIENTS,  MAXIMA  AND  MINIMA, 
ETC. 

We  have  already  passed  in  review  the  elementary  prin 
ciples  of  the  differential  calculus  to  a  liberal  and  compre 
hensive  extent.  But  we  have  purposely  deferred  geomet 
rical  illustration,  because,  if  we  had  hitherto  suffered  it  to 
engross  attention,  it  might  have  become  an  evil  so  great  as 
to  require  decisive  counteraction.  The  illusion  is  apt  to 
prevail  that  the  differential  calculus  relates  only  to  lines, 
or  forms ;  the  geometrical  construction  has  a  parallel  and 
independent  nature. 

236.  Every  algebraic  quantity  or  expression  not  imag 
inary,  consisting  of  an  aggregate  of  terms,  may,  by  per 
forming  the  algebraic  indications,  be  considered  resolved 
into,  or  constructed  as,  one  resultant  numerical  amount ; 
first  as  abstract  units,  inclusive  also  of  fractional  expressions 
of  units,  next  as  units  of  length,  such  as  may  be  rep 
resented  severally  as  each  a  straight  line,  or  collectively 
19 


218  DIFFERENTIAL  CALCULUS. 

as  a  continuous  straight  line.  Thus,  the  numerical  units, 
understood  to  be  intended  by  the  algebraic  quantity  a, 
may  be  represented  by  a  straight  line.  But  it  is  equally 

true  of  the  product  a  b,  of  the  quotient  -  ,  or  indeed  of 
such  an  aggregate  as 

a« 

"j~"T  Vc  a —  £3,  etc. 

Methods,  however,  are  pointed  out  in  analytical  geometry, 
by  which  the  values  of  Certain  expressions  may  be  illus 
trated  geometrically,  and  eliminating  all  considerations 
about  irrational  values,  by  the  methods  adopted  for  the 
construction. 

Next,  let  the  expression  contain,  be-  Y/         ._, 

sides  constants,  the  variable  &,  that  is, 
be  a  function  of  one  variable  x,  and  it 
becomes  evident  that  we  can  deter 
mine  a  straight  line  of  definite  length 
as  its  value,  by  supposing  a  value  for 
x.  Let  the  straight  line,  P  P'  (Fig.  1), 
represent  one  of  these  values,  and  let  it  have  the  more  per 
pendicular  position  for  a  conventional  reason,  its  universal 
adoption  by  geometers  to  represent  an  ordinate  of  a  plane 
curve,  or  of  such  line  as  may  be  determined  by  ordinates. 
This  leaves  a  chance  for  some  different  line,  I  P,  conven 
tionally  adopted  as  more  horizontal  to  be  the  record  of  the 
value  of  x,  and  they  have  the  point  of  meeting  P  in  com 
mon.  The  position  of  the  line  PP',  with  regard  to  the 
angle  it  makes  with  I  P,  may  be  any ;  it  may  be  perpen 
dicular  to  I  P,  but  need  not  be  necessarily  so.  All  these 
lines  and  points  are  to  be  supposed  to  be  in  one  plane. 

Next,  let  some  other  value  be  assumed  for  x,  and  for 
simplicity,  a  value  greater  by  a  small  amount  =  P  Q,  so 
that  we  now  have  x  =  I  Q,  and  let  the  corresponding  value 


GEOMETRICAL   ILLUSTRATIONS.  219 

of  Fx  be  deduced,  which  may  be  Q  Q',  and  let  it  be  par 
allel  to  P  P  ',  and  the  point  Q  be  taken  in  I  P  produced 
indefinitely  toward  X.  We  should  adopt  the  line  I X  as 
dividing  positive  values  of  F  x  from  negative  values  of  it. 
Let  those  values  of  Fx  on  the  side  of  I  X  toward  P',  Q' 
be  positive,  and  negative  values  of  F  x  will  be  found  on 
the  other  side  of  I  X.  Let  still  a  new  value  I  R  be  as 
sumed  for  cc,  and  a  corresponding  value  of  Fx  be  found 
and  be  represented  by  R  R  .  Indeed,  let  such  a  number 
of  values  of  x  be  assumed,  and  the  corresponding  values  of 
Fx  be  found  and  located,  sufficiently  near  together,  as  to 
give  a  complete  illustration  of  the  nature  of  these  succes 
sive  values. 

In  order  to  accommodate  such  conditions  as  grow  out 
of  x  at  negative  values,  I  being  the  origin  of  values,  and 
positive  being  conventionally  toward  the  righ%  X,  we  need 
the  line  I  X  extended  indefinitely  to  X '. 

Let  us  also,  through  I  at  any  angle  with  I  X,  draw  the 
straight  line  Y  Y'  as  an  original  line  of  indefinite  exten 
sion,  parallel  to  which  we  will  suppose  we  have  drawn 
PP',  QQ',  RR',  etc. 

The  intersecting  lines  XX'  and  Y  Y',  called  by  these 
designations  as  suggestive  of  the  values  of  x  measured  on  or 
parallel  with  the  former,  and  of  the  function  y,  measured 
on  or  parallel  with  the  latter,  are  the  same  as  the  axes  of 
coordinates  in  analytical  geometry.  We  may  call  them 
lines  of  reference,  which  are  to  be  supposed  existing  with 
reference  lo  all  the  constructions  of  the  values  of  functions 
of  one  independent  variable  we  may  wish  to  make  in  this 
section. 

Now,  the  continuous  line  which  shall  join  the  points  P7, 
Q',  R',  and  all  other  necessary  points  determined  in  the 
same  way,  is  the  particular  line  sought,  since  every  point 
in  it,  when  referred  to  Y  Y'  by  a  straight  line  parallel  with 
XX',  and  to  X  X'  by  a  straight  line  parallel  with  Y  Y', 


220 


DIFFERENTIAL   CALCULUS. 


shows  a  value  of  the  function  and  of  the  variable  in  cor 
respondence.  This  line  may  be  called  the  locus  of  the 
values  of  the  variables,  or  locus  of  the  equation. 

The  positions  of  the  four  conditions  alluded  to  in  Art. 
97,  become  quite  evident  in  the  construction. 

Now,  the  tediousness  of  this  way  of  proceeding  is  very 
much  relieved  by  the  adoption  of  certain  principles  de 
pending  on  the  character  of  the  function,  and  on  the  avail 
ability  of  differentiation. 


227.  The  construction  of  the 
values  of  a  function  of  a  variable 
derivable  from  or  referable  to  the 
general  equation  of  the  First  De- 
gree,  viz., 


is  simplest  by  making  x  =  0,  when  we  have  y  = » 

13 

which  value  set  off  on  I  Y'  determines  K  (Fig.  2) ;  making 

Q 

y  •=.  0,  we  derive  x  = ,  which  determines  the  negative 

A 

value  I  L,  for  x.     A  straight  line  through  K  and .  L,  ex 
tended  indefinitely,  both  ways,  is  the  line  sought. 

If  P  Q,  that  is  P '  M,  be  the  increment  h  of  the  variable, 
when  having  the  value  IP',  Q'M  becomes  the  decre 
ment  of  the  function  when  passing  the  value  PP'  (sup 
posing  P'  M  drawn  parallel  with  I  X),  that  is, 

F(x  +  h)  _Q'  M 
h  ~  P'M* 

and  this  quotient,  in  reference  to  the  equation  of  the  first 
degree,  happens  to  be  of  the  same  value  as 

dFx  dy 

— — ,  or,   - 


dx 


dx 


GEOMETRICAL   ILLUSTRATIONS.  221 

that  is,  as  the  differential  coefficient  of  the  function,  be 
cause  the  value  of  this  ratio  does  not  change  by  making 
A  =  0. 

As  examples,  lett  each  of  the  following  explicit  or  im 
plicit  functions  of  x  be  constructed  as  specified. 


3.  3(4£  +  2aO  +     =  0. 

4.  y  +  x  =  0. 

5.  ^_i_6  =  0. 
y 

228.  We  can  now  illustrate  geometrically  how  two 
algebraic  expressions,  each  containing  a  quantity  x  called 
unknown,  when  equated  with  each  other  as  in  the  solution 
of  some  algebraic  problems,  render  a  determinate  value 
for  #,  and  exclude  it  from  being  a  variable.  We  may 
construct  independently  each  of  these  expressions  as  a 
function  of  x.  The  point  of  a  common  value  of  each  of 
these  functions  determines  the  required  value  of  a?. 

This  principle  is  general  with  reference  to  equations  of 
different  degrees,  and  more  values  of  x  than  one.  At 
present,  however,  let  the  illustration  be  of  an  equation  of 
the  first  degree. 

Given  the  algebraic  equation  of  virtually  the  first  de 
gree,  viz., 

a  x  -(-  b  =  a'  x  -f-  #  ', 

to  construct  the  value  of  a?,  which  satisfies  the  condition, 
without  transposition  of  one  member. 

Let  I  M  (Fig.  3)  be  the  value  £,  that  is,  what  the  first 
member  becomes  when  x  =  0,  and  M  M'  be  the  locus  of 
all  values   of  a  x  +  6;  let  I  L  =  £',  and   L  L'  be  the 
19* 


DIFFERENTIAL   CALCULUS. 


Fig.  3. 


locus  of  all  values  of  a'  x  -(-  b'. 
If  they  intersect,  let  P'  be  the 
point  of  intersection ;  draw  P  P' 
parallel  to  Y  Y',  and  we  have 
I  P,  the  value  of  x  required.  If 
L  I/  and  M  M'  do  not  inter 
sect,  in  which  case  a  must  be 

a  =  a', 

x  will  be  indeterminate,  L  I/ 
M  M '  either  agreeing  or  being  parallel  to  each  other. 

If,  however,  transpositions  of  one  member  of  the  equa 
tion  first  take  place,  the  construction  will  be  one  straight 
line,  which  must  be  N  P,  I  N  being  equal  to  I  M  —  I  L  ; 
that  is,  to  b  —  b'  in 

(a  —  a')  x  -f  (b  —  b')  =  0 ; 
considering  a  ^>  a'  and  bf^>b. 

229.  In  the  arithmetical  rule  of  Double  Position,  in 
which  we  operate  without  the  possession  of  a  visible 
written  equation,  we  virtually  have,  in  the  conditions  of 
questions  ofiere-d  for  solution,  a  function  (referable  to  an 
equation  of  the  first  degree),  equal  to  zero,  to  find  JK,  and  we 
are  directed  to  suppose  any  two  numbers  for  the  unknown 
quantity,  and  to  test  each  of  them  in  the  conditions  with 
reference  to  finding  the  result,  zero ;  the  variations  from 
zero  we  call  errors  /  whence  by  the  use  of  the  supposed 
numbers,  and  the  errors,  we  derive  the  value  of  the  un 
known  quantity. 

In  Fig.  4,  let  N  P  be  con-      - 
struction  of 

A  x  +  B  —  0, 

to  find  x  or  I  P ;  we  suppose 
IP'=S,  to  be  IP,  and  find 
-the  error  P'  Q'  =  E;  next  Fig. 4. 


Q" 


GEOMETRICAL   ILLUSTRATIONS.  223 

we  will  suppose  I  P"  =  S'  to  be  IP,  and  find  the  error 
P"  Q"  =  E'.  Whence  we  have 

B:z::E±  E' : S' ±  S. 

This  is  an  algebraical  form  of  the  rule,  which,  in  arith 
metical  language,  is  necessarily  stated  with  much  circum 
locution. 

230,  It  is  evidently  impossible  to  draw  a  line  repre 
senting  the  differential  dx  of  a  variable  a?,  and  another 
representing  d  y,  the  differential  of  the  function  y,  each 
of  such  lines  being  zero  in  length ;  hence  we  cannot  ex 
hibit  in  visible  amounts  the  ratio  — .  But  we  can  exhibit 

d  x 

linear  amounts  of  which  the  value  of  their  ratio  is  exactly 
the  same  as  — .  Since  there  is  a  presumption  that  the 

d  x 

construction  of  the  values  of  functions  in  general,  of  a 
single  variable,  may  be  by  a  line  or  lines  not  necessarily 
straight,  but  curved,  although  without  regard  to  a  special 
function  we  can  determine  nothing  of  its  law,  let  a  part  of 
such  line  be  P"  P' S,  Fig.  5,  and  let  T  T '  be  a  straight  line 
meeting  it  at  P',  and  agreeing  with  it  in  the  nearest  vicin 
ity  ofP';  then, 

I  P  being  JG,  P  P'  being  y,   let 
PQ  =  P'  Q'  —  A,  and  we  have 
Q  ?"  =  F(x  +  h)  and  Q'P"= 
F  (x  -f-  A)  —  Fx  ;  hence  we  have 
F(x  +  A)  —  Fx      _P"Q 
h  ~P'Q'5 

and  when  h  =  0,  although  T'  Q',  P'Q'  each  become  0,  we 
have 

dy         T/  Q'         PP/ 


224  DIFFERENTIAL   CALCULUS. 

In  case  y  =  a  maximum  or  minimum,  we  evidently  have 
T  P  =  co,  T  T'  being  parallel  with  I  X,  so  that 


Now,  the  second  and  succeeding  dif.  coefs.  are  collectively 
represented  in  value,  by  the  ratio 

P//T/ 

P'   Q' 

in  the  value  it  assumes  when  P'  Q'  =  /i,  becomes  zero. 

With  a  view,  however,  to  make  originally  the  construc 
tion  of  a  given  function  of  a  single  variable,  and  to  deter 
mine  the  direction  of  the  line  P  '  Q'  R'  S'  (Fig.  1)  through 
any  point  P',  the  obvious  direction  is  to  determine  the 

value  of  T—  for  that  point,  and  by  this  value  construct  the 

d  x 

course  as  a  straight  line  for  the  immediate  vicinity  of  the 
point,  and  through  the  point. 

When  the  axes  of  reference  I  Y  and  I  X  are  rectangu 
lar,  the  value  of  —  for  any  point  P'  is  the  trigonometrical 

Cv  X 

tangent  of  the  angle,  which  the  straight  line  touching  the 
curve  at  that  point  makes  with  I  X,  called  the  axis  of  x  in 
'analytical  geometry. 

331.  Before  proceeding  to  the  geometrical  construc 
tion  of  equations  of  the  second  and  higher  degrees,  we 
properly  give  attention  to  illustrations  of  maxima  and 
minima  of  functions  of  one  independent  variable. 

The  character  of  a  maximum  is  shown 
in  Fig.  6,  by  observing  that  there  is  a  value 
at  P'  greater  than  the  nearest  contiguous 
values  on  either  side  of  it.  The  maximum  P 

value  of  the  function  is  P  P'.  Kg.  6. 


GEOMETRICAL   ILLUSTRATIONS. 


225 


A  minimum  is  shown  at  Fig.  7. 

A  maximum  and  minimum  of 
the  same  function  is  shown  in 
Fig.  8.  In  these  cases  —  is  sup 
posed  equal  to  zero. 

In  Fig.  9  is  shown  the  character 
of  a  maximum,  and  in  Fig.  10  of  a 

d  y 
minimum,  when  —  •=.  ±  &.  These 

d  x 

cases   evidently   come  within  the 
definitions. 


Fig.  7. 


Fig.  9. 


We   show,  in  Figs.  11 

and  12,  cases  illustrative  of 

—  =  0,  while  there  is  nei- 

d  x 

ther  a  maximum  nor  mini- 


P' 


P 

Fig.  11. 


mum.     Instances  by  actual  functions  are, 

by  Fig.  11,  y  =  a+(x  —  £)3, 

by  Fig.  12,  -  y  =  o—  (x  +  b)*. 

Another  instance   is  given   in   Fig.  13,  a 
corresponding  function  being 

y  —  b  +  (x  —  a)*. 


Fig.  8. 


Fig.  10. 


Fig.  12. 


P 

Fig.  13. 


232.   We  come  now  to  the  geometrical  con 
struction  of  the  general  equation  of  the  second  degree, 
which  is, 


(1.) 


B*        D 
whence  we  derive  y  =  —  o—  --  —  — 

2  A.          2  A. 


226  DIFFERENTIAL   CALCULUS. 

=b  —  I    /Y(B2_4AC)  tf2_j_2(BD 

2  A  I/ 

D2—4AF);  (2.) 

the  nature  of  which  may  be  written, 

y  =  A'x  +  'B'±  VCC'ajS  +  D's  +  E');         (2'.) 

from  (1.)  we  also  derive  x  =  --  -  —  — 

2  C        2C 


—  4CF);  (3.) 

the  nature  of  which  may  be  written, 

").       (3'.) 


Directing  our  attention  to  equation  (2.)  or  (2'.),  we  ob 
serve  that  its  radical 


may  have  the  value  zero,  at  some  value  or  two  values  of 
a?,  which  may  be  found  by  solving  the  equation 


these  values  found  and  substituted  for  x  in 

" 


(which  is  the  whole  equation  when  the  radical  is  zero), 
may  give  us  two  values  of  the  function  y  and  of  the  vari 
able  xy  which  must  belong  to  the  construction,  in  case  the 
values  are  not  imaginary. 


GEOMETRICAL  ILLUSTRATIONS.  227 

Now,  equation  (2".),  when  constructed,  must  be  repre 
sented  by  a  straight  line.     Let  L  L '  be  this  line,  and  the 


Fig.  14. 


points  P  and  P'  (Figs.  14  and  15),  be  determined  by  the 
values  of  x  just  found.     These  two  values  of  x  are, 


I 1 

C'     '4  C'2/ 

If  the  radical  term  of  this  value  of  x  is  neither  negative 
nor  zero,  we  certainly  have  two  values  of  x ;  if  zero,  we 
have  one  real  value ;  if  negative,  we  have  no  real  value, 
and  no  geometrical  construction  of  equation  (1.)  is  possible. 

We  will  suppose  we  have  two  real  values  of  x\  there 
fore  the  line  we  wish  to  construct  must  meet  the  straight 
line  L  L'  at  the  points  P  and  P'. 

Now,  by  differentiating  (2'.)  we  have,  when  the  radical 
equals  zero, 

*£==A,   .  IO/.  +  B' 


dx  2y(C'.ra-f-D>;r +  E') 

Therefore  the  line  sought,  and  necessarily  meeting  P  and 
P',  must  pass  through  these  points,  and  for  an  extremely 
short  distance  must  pass  through  them  parallel  with  I  Y, 
or  what  is  the  same,  I  Y '. 

We  will  now  proceed  to  regard  other  values  of  ic,  which 


228  DIFFERENTIAL   CALCULUS. 

revives  (2'.)  in  its  full  form,  and  will  consider  it  in 
three  respects;  ffrst,  when  C'  is  negative;  second,  when 
C  '  is  positive  ;  third  when  C  '  is  zero,  that  is,  when  in  (2.) 
or  (3.)  : 

(B2  —  4  AC)  <0; 

(B2_4  AC)  >0; 
(B2_4  AC)  ^0.  ' 

233,  When  (B  2  —  4  A  C)<  0.  Regarding  C  '  negative 
in  (2'.),  we  observe  that  the  values  of  x  must  evidently  be 
restricted  within  limits,  in  order  that  the  value  of  the 
radical  may  not  be  rendered  imaginary  (the  term  C'ce2 
being  the  ruling  term.)  This  is  so  whether  x  be  consid 
ered  positively  or  negatively  beyond  the  limits  already 
found  for  two  of  its  values,  P  and  P'.  Between  these  limits 
all  the  real  values  of  x  (and  consequently  y)  must  be  con 
tained  (Fig.  14). 

The  straight  line  L  I/,  already  located  and  produced  if 
necessary,  although  meeting  but  two  values  of  y,  is  never 
theless  one  from  which  all  the  values  of  the  radical  in  (2') 
are  to  be  set  off,  in  the  two  directions,  parallel  with  I  Y,' 
for  all  possible  values  of  y.  Taking  any  possible  value 
of  #,  we  may  substitute  it  for  x  in  the  radical,  and  deter 
mine  other  points  Q  and  Q',  R  and  R'  in  the  line  sought. 

But  it  will  be  more  interesting  and  expeditious  to  de 
termine  critical  or  singular  points.  Thus,  the  radical  of 


calling  it  a  separate  function  y',  must  have  its  own  max 
imum  and  minimum,  when 


2C 
by  which  value  of  ic,  we  determine  the  points  R  and  R', 


GEOMETRICAL   ILLUSTRATIONS.  229 

at  the  greatest  distance  from  LI/;  at  which  points  the  line 
sought  must  run  parallel  with  L  L  '.  Again,  y  has  its  max 
imum  and  minimum  separately  from  y'  .  Let  us  deter 
mine  them  at  P  and  P',  and  we  know  that  through  these 
points  the  line  sought  runs  parallel  with  I  X. 

We  have  already  discovered  sufficient  intimations  of  the 
nature  of  the  curve  line  we  are  endeavoring  to  trace  ;  that 
it  may  form  a  figure  called  an  ellipse,  but  possibly  a  circle, 
which,  however,  is  a  particular  case  of  an  ellipse.  The 
straight  line  P  P7  is  a  diameter,  and  must  bisect  all  straight 
lines  drawn  within  the  figure  parallel  to  I  Y'  and  termi 
nated  by  the  circumscribing  line. 

Precisely  the  same  result  should  we  have  arrived  at, 
had  we  originally  endeavored  to  construct  equation  (3.)  as  a 
function  of  y.  Its  form  is  the  same,  and  it  will  be  observed 
that  the  coefficient  of  y2,  under  the  radical,  viz.  (B2  —  4 
AC)  =  C',  is  the  same  in  both  (2.)  and  (3.),  which  should 
be  so  that  the  ellipse  should  be  equally  indicated  by  each 
for  the  condition  (B2  —  4  A  C)<  0.  But  instead  of  the 
line  LL',  we  should  have  II'  \  instead  of  I  Y,  we  should 
have  I  X';  instead  of  P  and  P',  we  should  have/*  and  />', 
and  any  lines  drawn  as  chords  in  the  figure  parallel  to 
IX  would  be  bisected  by  the  conjugate  diameter  p  pf, 

and  the  dif.  coefs.  —  and  —  ,  the  reciprocals  of  each  other 

d  x  dy 

in  value  for  any  given  point,  which  of  course  they  are  by 
notation. 

The  point  I'  of  the  intersection  of  these  diameters  may 
obviously  be  easily  found  at  the  outset  from  the  two 

equations, 

Ex       D 


where  we  have  virtually  two  (functions  of  x)  =  y,  and  a 
20 


230  DIFFERENTIAL   CALCULUS. 

common  possible  value  of  y  in  each,  as  also  of  se,  when 
the  value  of  x  and  y  at  the  intersection  1'  of  these  diame 

ters  will  be 

2AE—  BD 


(7.) 


Fig.  15. 

334,   When  B2  —  4  A  C  is  positive. 

Let,  as  before,  the  line  L  L/  (Fig.  15)  be  determined, 
since  the  construction  of  this  does  not  depend  on  the  sign 
ofB2  —  4  A  C. 

Let  also  certain  points,  P  and  P',  be  found,  at  which  the 
value  of  the  radical  becomes  zero,  and  for  each  of  which 

-  =  4;  GO  as  before.  Therefore,  through  P  and  P',  the  re- 

d  x 

quired  line  or  curve  will  pass  parallel  with  I  Y.  Let  us  next 
inquire  whether  it  is  between  these  two  values  of  x  now 
found  or  exterior  to  them,  that  all  the  real  values  of  y  are 
to  be  found,  and  we  may  at  once  suspect  that  it  is  exte 
rior,  for  the  reason  that  C'a;2,  (2'.)  being  positive,  the  rad- 
icfd  y'  may  have  real  values  of  unlimited  greatness,  posi 
tively  and  negatively,  as  well  as  ie,  while  between  these 


GEOMETRICAL   ILLUSTRATIONS.  231 

values  of  a-,  all  values  of  the  radical  part  of  y,  and  conse 
quently  y,  are  imaginary.  Indeed,  it  is  quite  evident  that 
when  x  is  sufficiently  small  or  restricted,  the  terms  D '  x  -f- 
E'  under  the  radical  become  ruling  terms,  over  C '  x%. 

Exterior  to  the  values  of  ic,  which  determine  P  and  P', 
let  any  value  of  x  be  assumed,  and  let  the  double  values 
of  the  radical  portion  of  y  be  found,  for  such  value  of  x ; 
this  will  determine  certain  points,  Q  and  Q',  R  and  R',  at 
equal  distances  from  L  I/,  as  real  values  of  the  function  y. 
In  this  manner  we  find  that  the  line  sought  will  be  a  curve 
consisting  of  two  infinite,  doubly  symmetrical  detached 
branches,  proceeding  in  opposite  directions.  It  is  the 
hyperbola. 

It  is  obvious  that  the  curve,  in  the  same  position  might 
have  been  constructed  from  equation  (3.),  as  the  figure 
will  show,  by  reading  the  previous  text  in  small  letters 
instead  of  capitals,  and  we  thus  determine  p  and  pr  etc. 
as  we  have  P  and  P',  etc. 

The  lines  P  P'  and  p  p'  are  called  diameters  of  the 
hyperbola.  If,  after  having  assumed  axes  of  reference,  we 
should  suppose  these  diameters  could  have  any  indepen 
dent  positions  under  the  general  equation  (1.),  we  should 
mistake,  for  their  positions  being  determined  by  equations 

d  *u 

(4.)  and  (5.)  are  such  that  — ,  from  each  of  these  two  virtual 

d  x 

functions  of  x  cannot  have  opposite  signs. 

Now,  evidently,  the  values  of  x  and  y  at  the  point  of  in 
tersection  of  these  diameters  for  the  hyperbola,  are  deter 
mined  by  equations  (6.)  and  (7.),  subject  to  regarding 
(B  2  —  4  A  C)  as  negative  for  the  ellipse,  and  positive  for 
the  hyperbola. 

235.   When  (B5*  —  4  A  C)  is  zero. 

In  this  condition  the  term  under  the  radical  in  (2.)  con 
taining  £c2  becomes  virtually  expunged. 


232  DIFFERENTIAL   CALCULUS. 

There  can  be  but  one  value  of  x  by  which  the  radical  in 
(2.)  can  have  the  value  zero. 

Let  the  line  L  L ',  Fig.  16,  be 
determined  as  in  the  two  pro-  ,,     , 

ceding  cases.  In  this  line  let 
the  one  point  P  be  found  at 
which  the  radical  equals  zero. 

As  before  —  for  this  value  is 
d  x 

zb  03,  indicating  as  before  the 
course  of  the  line  sought  while 

ITi  cr     1 C 

passing  through  P.     Consider 

next  whether  for  greater  or  for  less  values  of  x  than  that 
just  found  will  the  values  of  y  be  real.  If  greater,  deter 
mine  any  points,  Q  and  Q7  as  before,  and  any  other  points 
to  the  right  of  P.  This  curve  is  a  parabola,  of  which  L  L' 
is  a  diameter. 

We  should  have  constructed  the  same  figure  from  equa 
tion  (3.).  The  quantity  preceding  the  radical  would  de 
termine,  as  before,  some  other  line,  1 1'  parallel  with  L  L ', 
because  the  values  of  x  and  y  in  (6.)  and  (7.)  become  infi 
nite.  Then  we  should  find  the  location  of  p  in  the  same 
way  as  P,  q  and  q'  in  the  way  as  Q  and  Q'.  If  2  C  D  — 
B  E,  the  lines  L  L'  and  II'  become  one,  and  the  parabola 
becomes  merged  in  a  straight  line. 

In    order   to   construct   the   parabola   in 
the  position  required  to  represent  the  curve 
described    by    projected    bodies,   in    agree 
ment  with  the  natural  occurrence  (Fig.  17),       ' 
it  will  be  necessary  to  regard  the  axis  I  Y  Fl°- 17- 

as  perpendicular  to  the  horizon,  and  I X  as  horizontal,  and 
to  regard  in  equation  (1.)  A  =  0,  and  B  =  0,  which  is 
compatible,  as  it  must  be,  with  B2  —  4  AC  =  0.  Now, 
we  have  directly, 


GEOMETRICAL   ILLUSTRATIONS.  233 


In  order  to  obtain  the  same  value  of  y  from  equation 
(2.)  we  have  it  in  this  form, 


which,  if  its  meaning  be  sought,  must  be  found  equivalent 
to  the  previous  expression.  For  A  =  0,  and  B  =  0,  that 
reasoning  in  the  foregoing  text  fails,  which  required  a  value 
or  values  of  x  to  be  found,  at  which  the  radical  in  (2.)  be 
comes  zero,  since  it  cannot  become  so,  unless  D  —  0,  which 
it  need  not  be  ;  and  if  it  should  be,  we  must  have, 

0      0 

n  i     -     _     _  __ 

J  '  0       0  ' 

It  is  a  principle  that  no  line  of  the  second  order  can  be 
intersected  by  a  straight  line  in  more  than  two  points. 
Both  branches  of  the  hyperbola  can  be  considered  but  one 
line,  for  the  application  of  this  principle.  Curves  of  the 
second  degree  are  of  the  simplest  kind. 

We  have  now  given  summarily  the  general  construction 
of  equations  of  the  second  degree  ;  it  may  become  very 
much  simplified  for  particular  cases.  In  this  general  con 
struction  no  particular  regard  was  needed  to  the  particu 
lar  individual  signs  of  A  or  B,  etc. 

When  the  quantities,  of  whatever  kind  the  units  are 
that  enter  into  the  conditions  of  any  problem  in  this  trea 
tise,  can  be  placed  in  the  form  of  the  general  equation  of 
the  second  degree  between  two  variables,  the  relation  of 
the  real  values  of  this  implicit  function  of  either  of  them 
to  the  other  can  always  be  illustrated  by  the  construction 
of  the  indicated  curve,  or  straight  line  or  lines. 

These  curves  or  figures  are  the  same  as  those  produced 
by  the  plane  sections  of  a  cone,  or,  in  the  case  of  the  hy 
perbola,  of  two  similar  cones  of  infinite  axes  having  their 
20* 


234 


DIFFERENTIAL   CALCULUS. 


apexes  in  a  common  point,  and  their  axes  in  one  continu 
ous  straight  line.  The  section  of  the  plane  through  both 
cones  always  produces  the  hyperbola;  of  which  two  inter 
secting  straight  lines,  and  one  straight  line,  are  particular 
cases. 

A  section  of  one  of  the  cones  by  the  plane  which  if 
produced  will  not  meet  the  other,  if  finite,  produces  the 
ellipse,  if  infinite,  the  parabola. 

Fig.  18  is  a  construction  for  prob 
lem  27,  page  85. 

Fig.  19  is  a  construction  for 
4y2  —  20  yx  +  17  X*  —  0. 

Fig.  20  is  a  construction  for  prob 
lem  7,  page  122. 


Fig.  19.  Fig.  20. 

1.  Required  the  construction  of 

3  or       2     —  4=  x  —  3  =  0. 


2.  Required  the  construction  of  the  equation 

y2  —  2  ^  y  —  3  c«2  —  2y  +  7cc  —  1  =  0. 

3.  Required  the  construction  of  the  equation 

y2  —  4X2/_|_4a.2_|_2?/  —  1  x  —  1=0. 

4.  Required  to  find  the  roots  of  x  and  y,  which  in  the 
two  foregoing  equations  become  concurrent,  by  construct 
ing  both  equations  on  the  same  axes  of  reference. 


GEOMETRICAL   ILLUSTRATIONS.  235 

330.  It  would  be  impracticable,  within  the  limits  as 
signed  for  these  constructions,  to  undertake  that  of  the 
general  equation  of  the  third  or  any  higher  degree  be 
tween  two  variables  ;  but  we  may  make  selections  of  a 
few  functions  for  construction  without  regard  to  classifica 
tion  by  degrees,  and  with  reference  for  the  most  part  to 
particular  points  of  value,  and  to  the  vicinities  of  such 
points. 

In  the  construction  of  the  particular  case  of 


when  the  values  of  y  are  real,  there 

always  must  be  some  position  for  a 

straight   line   which  will   intersect 

the  curve   in   three   points.      The 

curve  must  pass  through  Y  Y'  (Fig. 

21),  at  the  value  D.     There  must 

be  a  point,  called  the  point  of  con-  Fio_ 

trary  flexure,  which   must    be    at 

the  value  of  a?  at  which 


and  the  curve  is  symmetrical,  with  reference  to  this  point, 
in  opposite  directions.  "Several  problems  in  a  previous 
section  are  based  on  a  function  of  x  of  this  nature. 


In  Fig.  22  is  given  the  complete  con 
struction  of 

2  —  a  x2  —  a;4. 


In  Fig.  23  is  given  the  construction, 
with  the  exception  of  infinite  values, 
for 

ic4  -(-  2  a  ic2  y  —  a  yz  =  0. 

Fig.  23. 


236 


DIFFERENTIAL   CALCULUS. 


In  Fig.  24  is  given  the  construction  for  x  =  a,  and  x  =  ft, 
and  for  values  of  &  between  a  and  ft,  and  greater  than  ft, 
for  the  equation, 

y  —  (x  —  a)*  X   V"»  —  b  +  c- 
In  Fig.  25  is  given  the  construction  of  the  equation, 


There  is  an  isolated  point  of  a  real  value  of  y,  at  y  =  c, 

^  y 
x  =  —  a  ;  at  this  point  —  is  imaginary  ;  which  indicates 

</  ./' 

that  the   curve  has  no  course  through  that  point.     The 
value  of  the  function  for  that  point  is  drawn. 


Fig.  24. 


Fig.  25. 


Fig.  26. 


In  Fig.  26  is  given  the  construction  for  x  —  0,   y  =  a 
and  the  vicinity,  for  the  equation, 


a. 


In  Fig.  27  is  given  the  construction  for 
one  point,  y  =  a  and  x  =  ft,  and  the  vi 
cinity,  for  the  equation, 

y  =  a  —  (x  —  ft)^  +  (%  —  ft)  • 

In  Fig.  28  is  given  the  construction, 
a  being  greater  than  ft,  for  the  points 
x  =  a,  x  =1  ft,  values  between  a  and  ft, 
and  greater  than  ft,  for  the  equation, 

y  =  (x  —  a)  x  V  x  — k 


Fig.  27. 


Fig.  28. 


GEOMETRICAL   ILLUSTRATIONS. 


237 


Let  the  function  of  x  for  problem  45,  Section  XII.,  be 
compared  with  this  last. 

237,  For  the  geometrical  illustration  of  the  value  of 
vanishing  fractions,  let  the  numerator  Fx  and  the  denom 
inator  fx  be  constructed  independently  on  the  same  axes 
of  reference,  for  the  value  zero  of  each  function,  and  for 
the  vicinity  of  that  value.  Let  N  P  N'  in  each  figure  be 
the  construction  for  the  numerator  F  x,  and  D  P  D'  be 
that  for  the  denominator  f  x,  the  point  P  being  that  of 
their  concurrent  value  zero.  Taking  P  O  =  A,  and  draw 
ing  S  O,  S  O,  or  S  S  '  parallel  with  I  Y,  we  shall  have 
for  that  value  of  x  next  succeeding  that  agreeing  with 
F  x  =  0,  and/a  =  0, 


F(x+h} 


OS 


As  demonstrated  in  the  section  on  the  subject,  we  have 
in  general  when  h  =  0, 

Fx       p'         p>' 

—  —  -  or,  -  -    or,  etc. 

fx         q'  q", 

1.   Fig.  29  exhibits  the  construction 
when  x  —  —  a,  for 


and  its  value  is,  by  — ,  =  -  ;    in  this 

case  it  is  obviously  no  matter  whether 
h  equal  zero  or  not. 


2.   Fig.  30  exhibits 


when  x  =  —  a,  its  value  be- 


238 


3.    Fig.  31  exhibits 
°  9' 


DIFFERENTIAL   CALCULUS. 

(a-*)' 


.  when  x  =  a,  its  value  be- 


Fig.  31. 


4.   Fig.  32  exhibits ,  when  x  =  a,  its   value   be- 

»"    "  (a  — a;)3 


Fig.  33. 


In  this  case,  since  both  first  dif.  coefs.  vanish,  the  inspec 
tion  of  the  figure  will  not  show  the  obviousness  of  the 
value  found,  unless  we  consider  that  as  h  becomes  0,  and 
each  first  dif.  coef.  0,  the  value  desired  may  be  considered 
without  a  more  particular  analysis,  as 


O  s'  — 0  S 
0  S  —0  S 


=  CO. 


GEOMETRICAL   ILLUSTRATIONS.  239 

238.  In  order  to  construct  geometrically  the  associated 
values  of  logarithms  and  their  natural  numbers,  let  the 
axes  of  reference  I  X  and  Y  Y'  intersect  at  I  and  at  any 
angle,  as  in  the  previous  constructions.  Since  the  loga 
rithmic  function  is 

y  —  log.  as, 

let  the  values  of  #,  that  is,  the  natural  numbers,  be  taken  in 
I  X  (Fig.  33),  commencing  at  I  ;  then  may  y,  that  is,  log.  x, 
be  taken  in  I  Y  when  positive,  and  in  I  Y  ',  when  negative. 
Take  some  distance  I  P  for  1,  and  since  log.  1  is  zero  for 
every  system,  the  required  line  will  meet  I  X  at  P.  Now, 
since  for  the  hyperbolic  system 


and  since  for  values  of  x  greater  than  1,  y  is  positive,  and 
for  values  of  x  less  than  1,  y  is  negative,  and  since  the  dif. 
coef.  has  but  one  value  when  x  —  1,  the  required  line 
will  pass  through  P,  equally  inclined  to  I  X  and  Y  Y'. 
Taking  Ip  =  l,=  modulus,  the  straight  line  joining 
p  P  will  be  tangent  to  the  curve,  showing  its  course 
through  P.  For  any  other  number  or  numerical  quantity, 
I  Q  or  I  R  determine  the  hyperbolic  logarithm  Q  Q',  or 

R  R',  and  by  the  corresponding  values  of  —  determine 

d  x 
the  course  of  the  curve  through  Q'  and  R'.     In  order  to 

find  where  the  tangents  for  any  points  Q'  and  R'  intersect 
Y  Y  ',  take  in  I  Y,  I  q  =  Q  Q  '  —  1,  1  r  =  R  R'  -  -  1,  and  q 
and  r  are  the  points,  respectively.  If  I  Q  =  2.71828,  Q  Q' 
=  2,  and  q  agrees  with  I.  If  I  R  =  7.38905  —  (2.71828)2, 
then  R  R'  =  2,  and  Ir  =  l.  A  line  passing  through 
Q',  R',  etc.,  wherever  they  may  be  determined,  as  also 
through  P,  is  the  logarithmic  curve  for  the  hyperbolic 
system. 

If  all  the  values  obtained  for  y,  by  the  method  just  given, 


240  DIFFERENTIAL   CALCULUS. 

be  divided  by  the  base  of  the  hyperbolic  system,  viz. 
2.71828,  the  values  of  y  for  the  common  system  will  be 
obtained.  For  this  system  the  modulus  being  .43429, 
this,  instead  of  1,  is  the  amount  by  which  any  logarithm 
must  exceed  the  value  from  I  on  Y  Y',  at  which  a  tangent 
to  the  curve,  corresponding  with  such  logarithm,  intersects 
Y  Y'.  This  curve  is.  drawn  in  the  figure.  If  I  S  =  10, 
then  S  &  =  1,  and  S  S'  =  2.30258  =  -^^-  —  the  ratio 
of  the  moduli. 

The  logarithmic  curve  may  sometimes  be  alluded  to  in 
books  when  rectangular  axes  of  reference  must  be  under 
stood  ;  as  in  the  case  of  the  area  of  such  curve. 


By  the  construction  of  all  functions  of  one  inde 
pendent  variable,  when  this  is  possible,  that  is,  when  the 
values  of  them  are  not  imaginary,  and  by  partial  construc 
tion  when  some  values  are  infinite,  we  exhaust  the  availa 
bility  of  a  plane  for  the  representation  ;  equally  so  whether 
the  axes  of  reference  be  rectangular  or  not. 

In  order  to  construct  a  function  *of  two  independent  va 
riables  notated  f  (x,  y)  •=.  z,  we  may  resort  to  space,  in 
which  solid  forms  are  embraced  ;  the  points,  lines,  planes, 
and  surfaces  defined  in  space,  however,  we  may  project  on 
a  plane,  as  by  Fig.  34. 

Let  the  plane  X  X'  Y  Y'  be  intersected  in  Y  Y' 
by  the  plane  Y  Y'  Z  Z',  and  also  in  X  X'  by  the  plane 
X  X'  Z  Z',  each  intersection  at  any  angle.  If  these 
angles  be  all  right  angles,  there  will  be  a  simplification 
of  the  principle  ;  but  it  should  be  borne  in  mind  that 
they  need  not  necessarily  be  so  for  all  purposes.  If  x  be 
taken  of  the  value  I  N,  and  y  of  the  value  I  O,  then  P  is 
the  point  at  which  these  values  concur;  for  these  val 
ues,  z  has  a  value  which  let  P  P'  parallel  with  Z  Z'  rep 
resent.  If  N  N  '  =  A,  then  P  P  '  becomes  Q  Q  '.  If  O  O  ' 
=  k,  so  that  I  O  '  =.  y  -\-  k,  then  for  x  =.  I  N,  z  becomes 


GEOMETRICAL   ILLUSTRATIONS. 


241 


R  R '.  So  that  for  y  -|-  £  and  for  x  -f-  h,  we  have  z  =  S  S' 
=  F  (x  +  A,  y  +  /O .  Through  P '  and  R '  let  a  line  P '  R ' 
in  the  plane  of  P  P',  R  R'  pass.  Through  Q'  and  S'  let 
a  line  Q'  S'  in  the  plane  of  Q  Q',  S  S'  pass.  Through  P' 
and  Q'  let  the  line  P'  Q'  in  the  plane  of  P  P7  and  Q  Q' 
pass.  Through  R'  and  S'  let  the  line  R'  S  in  the  plane 
of  R  R'  and  S  S'  pass.  The'  surface  expressed  by  P7  R', 
P'  Q',  R'  S',  Q'  S',  for  the  vicinity  of  P',  is  determinate 
by  the  positions  which  z  will  require  for  its  expression. 
The  function  F  (a-,  y)  =  2,  in  its  special  character,  must 
determine  the  law  of  this  surface,  for  the  positive  and  neg 
ative,  single  and  multiplied  values  of  z.  It  is  obvious  that 
the  variables  x  and  y  may  be  independent. 


If  attention  is  directed  only  to  the  plane  P  P'  Q  Q'. 

the  mode  of  representing   -  -   is  obvious ;    if  only  to   the 

d  x 

d  z 

plane  P  P'  R  R',  the  mode  of  representing  —  is  sufficient 
ly  obvious. 

The  surface  that  is  the  locus  of  the  general  equation  of 
the  first  degree  between  three  variables,  two  of  them  inde 
pendent,  the  equation  being 


21 


242  DIFFERENTIAL   CALCULUS. 

is  a  piano  any  how  posited  with  reference  to  the  three  as 
sumed  planes. 

The  general  equation  of  the  second   degree  between 
three  variables,  is 


H  y  +  K  x  +  L  =  0, 

the  construction  of  which  for  the  locus  of  2,  will  determine 
a  surface  of  such  a  character,  that  a  section  of  it  parallel 
with  one  of  the  planes  of  reference  produces  a  curve  of 
the  second  degree.  Such  a  surface  bounds  a  sphere;  or  a 
hyperboloid,  described  by  a  hyperbola  revolving  about  its 
transverse  axis  ;  —  or  a  paraboloid,  described  by  the  revo- 
tion  of  a  parabola  about  its  axis  ;  an  ellipsoid,  described  by 
the  revolution  of  an  ellipse  about  either  its  major  or  minor 
axis  ;  or  a  cone,  or  a  cylinder,  the  cone  being  a  particular 
case  of  the  hyperboloid  ;  either  of  them  any  how  posited. 

The  geometrical  construction  of  a  function  of  two  inde 
pendent  variables  of  a  higher  degree  than  the  second, 
will  of  course  be  by  surfaces,  but  we  can  do  no  more  than 
mention  the  methods  suggested  by  general  principles,  the 
determination  of  maxima  and  minima  for  z,  and  of  the 
position  of  the  surface  through  any  point  P'  by  the  values 

of  —  ,  —  ;  and  refer  to  the  analytical  geometry  of  three 

d  x   d  y 

dimensions,  in  systematic  treatises.     (J.  R.  Young's.) 

A  function  of  four  or  more  variables  can  have  no  geo 
metrical  construction  ;  nor  one  of  three,  when  the  value  of 
z  becomes  imaginary. 


Tn  closing  the  present  treatise,  it  is  deemed  worthy  of 
remark  that  at  least  three  American  treatises,  each  par- 


GEOMETRICAL   ILLUSTRATIONS.  243 

tially  devoted  to  the  differential  calculus,  present  as  Tay 
lor's  theorem,  both  in  the  enunciation  and  in  the  nature  of 
its  formula,  something  fatally  different  from  it,  and  of  an 
unwarranted  character.  Let  Young's  Differential  Calculus, 
one  of  the  most  scientific  in  the  English  language,  be  com 
pared  with  these ;  or  let  the  original  of  Brook  Taylor,  in 
the  Philosophical  Transactions,  be  consulted.  There  is  no 
need  of  argument  on  this  matter. 

In  one  of  the  three  treatises  there  is  an  apparent  success 
in  examples  employing  dx  as  the  equivalent  of  one  minute 
or  one  second  of  an  arc,  or  as  being  simply  one  in  pure 
number.  This  gives  a  result  that  for  the  number  of 
decimal  places  employed  in  the  trigonometrical  tables  may 
be  sufficiently  correct ;  simply  because  one  second,  or  even 
one  minute  happens,  by  the  arbitrary  division  of  the  arc 
of  a  circle  into  certain  number  of  degrees,  minutes,  or  sec 
onds,  a  matter  utterly  independent  of  the  calculus,  to  be 
so  small  an  arc.  If  the  number  of  decimal  places  were 
increased,  all  these  examples  would  be  found  to  fail. 

But  having  the  logarithm  of  any  number  a  to  find  log. 
(a  -f-  1)  in  the  manner  adopted  in  the  same  treatise,  by 
adding  to  log.  a  simply  — ,  is  to  ignore  the  terms  that  fol- 

a 

low  in  the  brackets,  where  we  give  the  full  development 
for  the  hyperbolic  system,  of  the  direct  point  at  issue,  viz., 

the  number  a  must  be  very  large  that  this  method  may 
not  notoriously  fail,  for  even  the  early  decimal  places  of 
the  logarithmic  tables.  Yet  we  find  no  intimation  of  a 
caution. 

It  is  plain  that  dx  cannot  be  anything  in  value  else 
than  0,  since,  by  hypothesis  and  by  definition,  it  is  made  so. 

One  of  these  treatises  (Art.  13)  undertakes  to  demon- 


244  DIFFERENTIAL   CALCULUS. 

strate  that  "  if  two  functions  are  equal,  their  differentials  are 
also  equal ;"  an  impossible  general  truth,  considering  these 
differentials  as  zeros,  but  necessarily  independent.  The 
apparent  demonstration  is  a  tautology  of  notation. 

The  same  treatise,  in  Art.  37,  undertakes  to  differentiate 
log.  v  by  assuming,  in  the  course  of  the  demonstration, 
the  development  of  log.  (1  -|-  y),  a  development  which 
must  depend  upon  the  differentiation  of  log.  (1  -f-  y). 
Hence,  the  necessary  analysis  of  differentiating  log.  v  is  all 
eliminated,  and  absent,  and  the  demonstration  void.  Op 
portunities  requiring  criticism,  as  well  taken  as  in  the 
instances  we  have  just  presented,  are  copious  throughout 
the  part  devoted  to  the  differential  calculus. 


THTC    END. 


HJ 


Page  15,  line  7,/or  ^  being,  read  f  and  x. 
P.  31,  at  foot,  dele  —  h  from  the  minuendive  fraction. 
P.  45,  in  example  8,  for  3W,  read  3  n. 
P.  68,  1.  16,yor  the  second  —  h,  read  -\-  h. 
P.  73,  ex.  17.  for  the  exponent  |,  read  f. 
P.  98,  1.  3,  after  range,  read  of-the  product  of  any  two 
20,  1.  2,/or  —  3  —  i  or  Tfo,  read  —  |  or  —  3.08. 
P.  140,  1.  3,  for  gained,  read  lost. 
P.  192,  equation  2.  rfe/e  dx  from  the  numerator. 


P.  201,  1.  6,  draw  a  mnculum  over  L  (T 

P.   236,   1.   6,  for  7/2   IB  Xfa  +  xj2    +   bj  rcad  y  = 

&  ( a  +  x)  -f  c. 
P.  244,  dele  lines  5th  to  the  1  \th. 


14  DAY  USE 

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